MULTIVARIABLE ORTHOGONAL POLYNOMIALS AND COUPLING COEFFICIENTS FOR DISCRETE SERIES REPRESENTATIONS

HJALMAR ROSENGREN

Abstract. We study polynomials of several variables which occur as coupling coefficients for the analytic continuation of the holomorphic discrete series of SU(1, 1). There are three types of such polynomials, one corresponding to each conjugacy class of one-parameter subgroups. They may be viewed as multivariable generalizations of Hahn, Jacobi, and continuous Hahn polynomials and include many orthogonal and biorthogonal families occurring in the literature. We give a simple and unified approach to these polynomials using the group theoretic interpretation. We prove many formal properties, in particular a number of convolution and linearization formulas. We also develop the corresponding theory for the Heisenberg group, leading to multivariable generalizations of Krawtchouk and Hermite polynomials.

1. Introduction

In [R], we introduced three classes of polynomials in several variables. They appeared there as coupling coefficients for the group SU(1, 1) ≃ SL(2, ℝ) and its covering groups. Special cases occur in the work of many authors, cf. [A1], [A2], [AK], [E], [Ex], [FL], [KMT], [KM1], [KM2], [K1], [KS], [LT], [M], [MP], [Pr], [Ra], [T1], [T2], [T4], [T5], and [V]. In this paper we give a simple and unified approach to the study of these polynomials. We also consider the case of the oscillator or Heisenberg group, leading to two more classes of polynomials.

In many cases such multivariable polynomials arose naturally from problems in physics or statistics. In view of the group theoretic interpretation this is not surprising, since symmetry groups play an important role in these sciences. There are also close connections with spherical harmonics, as well as with the Wigner symbols (recoupling coefficients) occurring in quantum mechanics.

We will work with the analytic continuation of the holomorphic discrete series of SU(1, 1) ≃ SL(2, ℝ). We realize these representations on the Hilbert spaces A^{ν} (ν > 0) of analytic functions on the complex unit disc, with the scalar product

⟨f, g⟩ = ∑_{k=0}^{∞} \frac{k!}{(ν)_{k}} f̂(k)\overline{ĝ(k)},

where f(z) = ∑ f̂(k)z^{k} and

(ν)_{k} = \frac{Γ(ν + k)}{Γ(ν)} = ν(ν + 1) ⋯ (ν + k − 1)

1991 Mathematics Subject Classification. 33C50, 33C55, 33C80. Key words and phrases. multivariable orthogonal polynomials, multivariable biorthogonal polynomials, coupling

coefficient, spherical harmonics, multivariable hypergeometric series, convolution formula, linearization formula.

This paper was written in 1997, while the author was employed by Lund University. It forms part of the author’s PhD thesis, written under the guidance of Professor Jaak Peetre. It has appeared in print: SIAM J. Math. Anal. 30 (1999), 233–272.

is the Pochhammer symbol. The norm is invariant under the transformations

f(z) ↦ f (\frac{αz + β}{γz + δ} ) \frac{1}{(γz + δ)^{ν}} , (\begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ SU(1, 1).

This may be used to define a unitary representation of the universal covering group of SU(1, 1) on each space A^{ν}, cf. [Sa] for the details.

We consider a Hilbert tensor product A^{ν_{1}} ⊗⋯⊗A^{ν_{n}} of such spaces. It decomposes under the group action as

(1)  A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} ≃ ⨁_{s=0}^{∞} \binom{n + s − 2}{n − 2}A^{|ν|+2s}

(where |ν| = ∑ ν_{i}). A highest weight vector Q is an image of the constant function 1 under an intertwining embedding

(2)  A^{|ν|+2s} → A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}}.

In our realization, this means that Q is a homogeneous polynomial of degree s, which may be expressed as a function of differences z_{i} − z_{j} of the coordinates. Equivalently, Q satisfies the identity

(3)  Q(az_{1} + b, …, az_{n} + b) = a^{s}Q(z_{1}, …, z_{n}), a, b ∈ ℂ.

In agreement with (1), the space of such polynomials has dimension \binom{n+s−2}{n−2}.

In [R], we introduced three transforms which we will denote here by T_{1}, T_{2}, and T_{3}. Their role is to send highest weight vectors to coupling coefficients. They are defined on z^{k} ∈ A^{ν} by

(4)  T_{1}z^{k}(m) = (−1)^{k} \frac{(−m)_{k}}{(ν)_{k}} , T_{2}z^{k}(ξ) = \frac{ξ^{k}}{(ν)_{k}} , T_{3}z^{k}(X) = (−1)^{k} \frac{(\frac{ν}{2} − iX)_{k}}{(ν)_{k}}

and extended to power series by linearity. We will also denote by T_{i} the extension T_{i} ⊗ ⋯ ⊗ T_{i} to a tensor product A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}}. Thus if we let

Q(z) = ∑_{|t|=s} c_{t} z_{1}^{t_{1}} ⋯ z_{n}^{t_{n}}

be a highest weight vector in A^{ν_{1}}⊗⋯⊗A^{ν_{n}} (throughout the paper we use multi-index notation freely), the three transforms of Q are given by

(5)  T_{1}Q(m_{1}, …, m_{n}) = (−1)^{s} ∑_{|t|=s} c_{t} \frac{(−m_{1})_{t_{1}} ⋯ (−m_{n})_{t_{n}}}{(ν_{1})_{t_{1}} ⋯ (ν_{n})_{t_{n}}} ,

(6)  T_{2}Q(ξ_{1}, …, ξ_{n}) = ∑_{|t|=s} c_{t} \frac{ξ_{1}^{t_{1}} ⋯ ξ_{n}^{t_{n}}}{(ν_{1})_{t_{1}} ⋯ (ν_{n})_{t_{n}}} ,

(7)  T_{3}Q(X_{1}, …, X_{n}) = (−1)^{s} ∑_{|t|=s} c_{t} \frac{(\frac{ν_{1}}{2} − iX_{1})_{t_{1}} ⋯ (\frac{ν_{n}}{2} − iX_{n})_{t_{n}}}{(ν_{1})_{t_{1}} ⋯ (ν_{n})_{t_{n}}} .

Note that the T_{i}Q are again polynomials of degree s, though only T_{2}Q is homogeneous. Moreover, T_{1} and T_{3} are connected by the equation

(8)  T_{3}Q(X) = T_{1}Q (iX − \frac{ν}{2}) .

One may also prove that

(9)  T_{3}Q(X) = (−1)^{s} T_{3}Q(−X),

which is not obvious from the definition.

In [R] we proved that if Q and Q′ are two highest weight vectors, then the quantities

∑_{m_{1}+···+m_{n}=M}\frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(m_{1}, …, m_{n}) \overline{T_{1}Q′(m_{1}, …, m_{n})}

for M = 0, 1, 2, …,

∫_{ξ∈ℝ_{+}^{n}: |ξ|=C} T_{2}Q(ξ) \overline{T_{2}Q′(ξ)} ξ_{1}^{ν_{1}−1} ⋯ ξ_{n}^{ν_{n}−1} dξ,

where |ξ| = ξ_{1} + ⋯ + ξ_{n} and C > 0 is arbitrary, and

∫_{X∈ℝ^{n}: |X|=C} T_{3}Q(X) \overline{T_{3}Q′(X)} |Γ (\frac{ν_{1}}{2} + iX_{1})|^{2} ⋯ |Γ (\frac{ν_{n}}{2} + iX_{n})|^{2} dX,

where C ∈ ℝ is arbitrary, are all proportional to the scalar product

⟨Q, Q′⟩_{A^{ν_{1}}⊗···⊗A^{νn}}.

In particular, given an orthogonal basis for the highest weight vectors, our transforms give us three systems of orthogonal polynomials. The form of the orthogonality relations suggests that one eliminate one variable and consider, for instance, for fixed M,

T_{1}Q(m_{1}, …, m_{n−1}, M − m_{1} − ⋯ − m_{n−1})

as a polynomial in n − 1 variables. This is how these polynomials usually occur in the literature.

In the case when all the ν_{i} are half-integers, there is a connection with spherical harmonics. If we introduce the polynomial

R(x_{1}, …, x_{2|ν|}) = (T_{2}Q)(x_{1}^{2} + ⋯ + x_{2ν_{1}}^{2}, …, x_{2(ν_{1}+···+ν_{n−1})+1}^{2} + ⋯ + x_{2|ν|}^{2}),

then the orthogonality relations for T_{2}Q give corresponding orthogonality relations for R on the unit sphere of ℝ^{2ν}. Moreover, the conditions on Q ensure that R is harmonic. In fact one obtains precisely the space of harmonic polynomials of degree s invariant with respect to the subgroup

O(2ν_{1}) × ⋯ × O(2ν_{n}) ⊆ O(2|ν|),

called polyspherical harmonics [V]. In [R] we gave an explanation of this, which involved the Howe dual pair Mp(1) × O(k) ⊆ Mp(k).

The plan of the paper is as follows. In Section 2 we describe some special cases of our polynomials, in particular those that we have found in the literature. In Section 3 we review some fundamental facts on matrix elements of our representation due to Basu and Wolf [BW1]. In Section 4 we obtain the polynomials T_{i}Q as coupling coefficients for our representation. The proof is based on certain factorizations of the Fourier transforms on A^{ν} with respect to one-parameter subgroups. This is different from the approach in [R]. In Section 5 we prove convolution formulas for our polynomials. These generalize formulas recently found by Koelink and Van der Jeugt [KV1]. In Section 6 we introduce some polynomials in 2n variables which we call coupling kernels. Whereas the coupling coefficients T_{i}Q are connected with matrix elements of the intertwining embeddings (2) (or of the corresponding projections A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} → A^{|ν|+2s}), the coupling kernels are connected with matrix elements for the projection of A^{ν_{1}} ⊗⋯⊗A^{ν_{n}} onto the isotypic subspace equivalent to \binom{n+s−2}{n−2}A^{|ν|+2s} in (1). In Section 7 we study formal properties of the coupling kernels. These include some linearization formulas, which generalize the classical Burchnall– Chaundy formula. In Section 8 we prove that the coupling kernels may be expressed in terms of certain generalized hypergeometric functions. Finally, in Section 9 we give the analogues of our results when SU(1, 1) is replaced by the Heisenberg group and the spaces A^{ν} are replaced by Fock spaces of entire functions.

2. Special cases

We will now describe some special cases of our polynomials. We have found many occurrences of them in the literature after writing [R], so we will give more complete references here. We first recall the definition of the generalized hypergeometric function

pF_{q} ( \begin{matrix}a_{1}, …, a_{p} \\ b_{1}, …, b_{q} \end{matrix} | x) = ∑_{k=0}^{∞} \frac{(a_{1})_{k} ⋯ (a_{p})_{k} x^{k}}{(b_{1})_{k} ⋯ (b_{q})_{k} k!} .

When a and k are vectors, we write

(a)_{k} = (a_{1})_{k_{1}} ⋯ (a_{n})_{k_{n}}, k! = k_{1}! ⋯ k_{n}!.

We will also encounter Karlsson’s generalized Kampé de Fériet functions [Ka], [SK], which we may then write, with a slight variation of Karlsson’s notation, as

F_{r:s}^{p:q} ( \begin{matrix}a_{1}, …, a_{p} & : & b_{1}, …, b_{q} \\ c_{1}, …, c_{r} & : & d_{1}, …, d_{s} \end{matrix} | x)

(10)

= ∑_{k_{1},...,k_{n}=0}^{∞} \frac{(a_{1})_{|k|} ⋯ (a_{p})_{|k|}(b_{1})_{k} ⋯ (b_{q})_{k} x^{k}}{(c_{1})_{|k|} ⋯ (c_{r})_{|k|}(d_{1})_{k} ⋯ (d_{s})_{k} k!} ,

where the a_{i} and c_{i} are scalar parameters while the b_{i}, d_{i}, and x are vectors of the same dimension n. For n = 1, F_{r:s}^{p:q} reduces to the function _{p+q}F_{r+s}.

2.1. The bilinear case. The simplest case is when n = 2. Then the space of highest weight vectors of each degree s is one-dimensional and spanned by the single element

Q(z_{1}, z_{2}) = (z_{1} − z_{2})^{s}.

In this case, the polynomials T_{1}Q are Clebsch–Gordan coefficients, which may be expressed in terms of Hahn polynomials. In fact,

T_{1}Q(m_{1}, m_{2}) = ∑_{k=0}^{s} \binom{s}{k}(−1)^{k} \frac{(−m_{1})_{k}(−m_{2})_{s−k}}{(ν_{1})_{k}(ν_{2})_{s−k}}

= \frac{(−m_{1} − m_{2})_{s}}{(ν_{2})_{s}} Q_{s}(m_{1}; ν_{1} − 1, ν_{2} − 1; m_{1} + m_{2}),

where Q_{s} is the Hahn polynomial of degree s in the notation of [VK]. The polynomials T_{2}Q and T_{3}Q are Clebsch–Gordan coefficients with respect to continuous bases; they may be expressed in terms of Jacobi polynomials and “Hahn polynomials of an imaginary argument,” respectively; cf. [MR], [BW2], compare also [P2] and [VK], Section 8.7. Still in the notation of [VK], we have

T_{2}Q(ξ_{1}, ξ_{2}) = ∑_{k=0}^{s} \binom{s}{k}(−1)^{s−k} \frac{ξ_{1}^{k} ξ_{2}^{s−k}}{(ν_{1})_{k}(ν_{2})_{s−k}}

= \frac{(−1)^{s}s!}{(ν_{1})_{s}(ν_{2})_{s}} (ξ_{1} + ξ_{2})^{s}P_{s}^{(ν_{1}−1,ν_{2}−1)} (\frac{ξ_{2} − ξ_{1}}{ξ_{1} + ξ_{2}} ) ,

T_{3}Q(X_{1}, X_{2}) = ∑_{k=0}^{s} \binom{s}{k}(−1)^{k} \frac{(\frac{ν_{1}}{2} − iX_{1})_{k}(\frac{ν_{2}}{2} − iX_{2})_{s−k}}{(ν_{1})_{k}(ν_{2})_{s−k}}

= \frac{(\frac{ν_{1}+ν_{2}}{2} − i(X_{1} + X_{2}))_{s}}{(ν_{2})_{s}} q_{s}(X_{1}; \frac{ν_{1}}{2} , \frac{ν_{2}}{2} − i(X_{1} + X_{2})).

Even in this case, the present paper gives new and simple deductions of properties of these polynomials.

2.2. Polynomials of Appell-type. In the case of general n, the polynomials

(11)  Q_{t}(z) = (z_{1} − z_{n})^{t_{1}} ⋯ (z_{n−1} − z_{n})^{t_{n−1}}, t_{1} + ⋯ + t_{n−1} = s

form a basis for the space of highest weight vectors of degree s. Expanding Q_{t} by the binomial theorem, one finds that the polynomials T_{i}Q_{t} may be expressed in terms of Karlsson’s generalized Kampé de Fériet functions (10) with the vector parameters of dimension n − 1. In fact, one has

T_{1}Q_{t}(m_{1}, …, m_{n})

\begin{matrix}= \frac{(−m_{n})_{s}}{(ν_{n})_{s}} ∑_{k_{1},...,k_{n−1}=0}^{∞} \frac{(1 − ν_{n} − s)_{|k|}(−t_{1})_{k_{1}}⋯ (−t_{n−1})_{k_{n−1}}(−m_{1})_{k_{1}} ⋯ (−m_{n−1})_{k_{n−1}}}{(1 + m_{n} − s)_{|k|}(ν_{1})_{k_{1}} ⋯ (ν_{n−1})_{k_{n−1}}k_{1}! ⋯ k_{n−1}!} \\ = \frac{(−m_{n})_{s}}{(ν_{n})_{s}} F_{1:1}^{1:2} ( \begin{matrix}1 − ν_{n} − s & : & −t, −(m_{1}, …, m_{n−1}) \\ 1 + m_{n} − s & : & (ν_{1}, …, ν_{n−1}) \end{matrix} | 1) \end{matrix}

(where, as is customary, we write the variable of the F_{1:1}^{1:2}-function as 1 = (1, …, 1)). By (8), a similar expression is valid for T_{3}Q_{t}, while T_{2}Q_{t} is given by

T_{2}Q_{t}(ξ_{1}, …, ξ_{n}) (12)

= \frac{(−ξ_{n})^{s}}{(ν_{n})_{s}} F_{0:1}^{1:1} ( \begin{matrix}1 − ν_{n} − s & : & −t \\ − & : & (ν_{1}, …, ν_{n−1}) \end{matrix} | − \frac{ξ_{1}}{ξ_{n}} , …, −\frac{ξ_{n−1}}{ξ_{n}} )

(Karlsson’s F_{0:1}^{1:1}-function is also known as the Lauricella F_{A}-function).

2.3. Polynomials biorthogonal to the polynomials of Appell-type. The family (11) is obviously nonorthogonal for n ≥ 3. Thus it is a problem to find the dual basis. Applying the T_{i} to these two bases then gives biorthogonal polynomial systems. Though we have found no very simple expression for the basis dual to (11), the polynomials obtained after applying the T_{i} are also expressible in terms of Karlsson’s functions; cf. Corollary 6.2 below.

The polynomials (12) were introduced by Appell [A1] for n = 3. The dual basis was found by him [A2] when ν_{3} = 1 and by Engelis [E] and, independently, by Fackerell and Littler [FL] in general. (For the work of Appell, see also the book [AK].) Lam and Tratnik [LT] generalized these polynomials to general n. The corresponding spherical harmonics occur in [KMT]. In the special case when all the ν_{i} equal 1, these harmonics go back to Hermite, Didon, and Kampé de Fériet; cf. again [AK]. The biorthogonal systems obtained using T_{1} were introduced by Rahman [Ra] for n = 3 and by Tratnik [T2] in general. For the transform T_{3}, the corresponding polynomials are also due to Tratnik [T1].

2.4. Polynomials of Proriol-type. There seems to be no canonical choice of an orthogonal basis in the space of highest weight vectors. There is, however, a general procedure, known as binary coupling (the coupling of angular momenta in quantum physics), that allows one to construct many such bases. The corresponding polynomials T_{i}Q are given by products of n − 1 factors, each factor being a Clebsch– Gordan coefficient, that is, a Hahn, Jacobi, or continuous Hahn polynomial.

As an example, let us consider the case n = 3. Let K_{s}^{ν_{1},ν_{2}} denote the intertwining embedding

A^{ν_{1}+ν_{2}+2s} → A^{ν_{1}} ⊗ A^{ν_{2}}

such that K_{s}^{ν_{1},ν_{2}}1 = (z_{1} − z_{2})^{s}. Then the polynomials

Q_{st} = (K_{s}^{ν_{1},ν_{2}} ⊗ id) ∘ K_{t}^{ν_{1}+ν_{2}+2s,ν_{3}}1, s + t = N

form an orthogonal basis for the space of highest weight vectors in A^{ν_{1}} ⊗ A^{ν_{2}} ⊗ A^{ν_{3}} of degree N. In [R] we found that

Q_{st}(z) = (z_{1} − z_{2})^{s} ∑_{i+j=t} \frac{t!}{i! j!} \frac{(ν_{1} + s)_{i}(ν_{2} + s)_{j}}{(ν_{1} + ν_{2} + 2s)_{t}} (z_{1} − z_{3})^{i}(z_{2} − z_{3})^{j},

T_{1}Q_{st}(m) = \frac{(−m_{1} − m_{2})_{s}(s − m_{1} − m_{2} − m_{3})_{t}}{(ν_{2})_{s}(ν_{3})_{t}} Q_{s}(m_{1}; ν_{1} − 1, ν_{2} − 1; m_{1} + m_{2})

×Q_{t}(m_{1} + m_{2} − s; ν_{1} + ν_{2} + 2s − 1, ν_{3} − 1; m_{1} + m_{2} + m_{3} − s),

T_{2}Q_{st}(ξ) = \frac{(−1)^{s+t}s! t!(ξ_{1} + ξ_{2})^{s}(ξ_{1} + ξ_{2} + ξ_{3})^{t}}{(ν_{1})_{s}(ν_{2})_{s}(ν_{1} + ν_{2} + 2s)_{t}(ν_{3})_{t}}

×P_{s}^{(ν_{1}−1,ν_{2}−1)} (\frac{ξ_{2} − ξ_{1}}{ξ_{1} + ξ_{2}} ) P_{t}^{(ν_{1}+ν_{2}+2s−1,ν_{3}−1)} (\frac{ξ_{3} − ξ_{1} − ξ_{2}}{ξ_{1} + ξ_{2} + ξ_{3}} ) ,

T_{3}Q_{st}(X) = \frac{(\frac{ν_{1}+ν_{2}}{2} − i(X_{1} + X_{2}))_{s}(\frac{ν_{1}+ν_{2}+ν_{3}+2s}{2} − i(X_{1} + X_{2} + X_{3}))_{t}}{(ν_{2})_{s}(ν_{3})_{t}}

×q_{s} (X_{1}; \frac{ν_{1}}{2} , \frac{ν_{2}}{2} − i(X_{1} + X_{2}))

×q_{t} (X_{1} + X_{2}; \frac{ν_{1} + ν_{2} + 2s}{2}, \frac{ν_{3}}{2} − i(X_{1} + X_{2} + X_{3})) .

The polynomials T_{2}Q_{st}, with Q_{st} as above, were introduced by Proriol in [Pr]. They were independently obtained and applied to genetics by Karlin and McGregor in [KM1] and have also found applications in physics [M], [MP]. See the survey [K1] for these and other two-variable analogues of Jacobi polynomials. The spherical harmonics constructible by this method were, for general n, introduced in [V], cf. also [VK], Section 10.5, and generalized further in [KMT]. For the special coupling procedure

Q_{s_{1},...,s_{n−1}} = (K_{s_{1}}^{ν_{1},ν_{2}} ⊗ id) ∘ (K_{s_{2}}^{ν_{1}+ν_{2}+2s_{1},ν_{3}} ⊗ id) ∘ ⋯ ∘ K_{s_{n−1}}^{ν_{1}+···+ν_{n−1}+2(s_{1}+···+s_{n−2}),ν_{n}}1,

the polynomials T_{1}Q_{s_{1},...,s_{n−1}} were defined by Karlin and McGregor in [KM2], cf. also [T5]; again they appear in the context of genetics. The corresponding polynomials T_{3}Q_{s_{1},...,s_{n−1}} were introduced by Tratnik in [T4]. An interesting product formula for the polynomials T_{2}Q_{s_{1},...,s_{n−1}} is given in [KS].

3. Matrix elements

In this section we write down some basic facts, due to Basu and Wolf [BW1], cf. also [K2], about matrix elements of our representation. Whereas these authors work with the oscillator realization on L^{2}(ℝ_{+}), we will reformulate their results in terms of the analytic function spaces A^{ν}. This has computational advantages, allowing us to work with power series expansions.

Let D be the unit disc of ℂ and G ≃ SU(1, 1)/{±1} be the group of conformal automorphisms of D. There are three conjugacy classes of one-parameter subgroups in G. They may be described in terms of the fixed point sets in ℂ ∪ {∞} of the corresponding Möbius maps. There are the elliptic elements, which fix one point in D and one in the outside, the hyperbolic elements, which fix two points on the boundary of D, and finally the parabolic elements, which have a single fixed point on the boundary.

We choose a representative subgroup in each conjugacy class, namely, the elliptic subgroup

ϕ_{t}^{1} = (\begin{matrix}e^{it/2} & 0 \\ 0 & e^{−it/2} \end{matrix}) ,

corresponding to rotations z ↦ e^{it}z of the disc, the parabolic subgroup

ϕ_{t}^{2} = (\begin{matrix}1 + it/2 & it/2 \\ −it/2 & 1 − it/2 \end{matrix}) ,

and the hyperbolic subgroup

ϕ_{t}^{3} = (\begin{matrix}cosh(t/2) & − sinh(t/2) \\ − sinh(t/2) & cosh(t/2) \end{matrix}) .

One may check that

ϕ_{s}^{i}ϕ_{t}^{i} = ϕ_{s+t}^{i}, i = 1, 2, 3.

Note that if we choose ψ as the Cayley transform

ψ(z) = \frac{i − z}{i + z}

of the upper half-plane onto D, then

(ψ^{−1}ϕ_{t}^{2}ψ)(z) = z + t,

so we may realize the parabolic subgroup as translations of the half-plane. Similarly

(γ^{−1}ϕ_{t}^{3}γ)(z) = z + t,

where

γ(z) = \frac{i − e^{z}}{i + e^{z}} ,

which realizes the hyperbolic subgroup as translations of the strip {z | 0 < Imz < π}. (Hyperbolic transformations may also be realized as dilatations of the half-plane.)

To clarify the group theoretic meaning of our formulas, we employ certain notational conventions, which we will now state explicitly. Various objects connected with the subgroup ϕ_{t}^{i} (i = 1, 2, 3) will be labelled with the index i. The letter z will denote variables ranging in the disc (or polydisc). We will be concerned with three versions of the Fourier transform, one for each subgroup. For the elliptic subgroup, the variable dual to z is discrete; it will be denoted by k, l, m. For the parabolic and hyperbolic subgroup, the dual variables are continuous; they will be denoted by ξ, η and X, Y, respectively. The equations (5), (6), and (7) exemplify these conventions.

Recall now the function spaces A^{ν} (ν > 0) with scalar product

⟨f, g⟩ = ∑_{k=0}^{∞} \frac{k!}{(ν)_{k}} f̂(k)\overline{ĝ(k)}.

We will use this notation also when f or g is not in A^{ν}, provided that the sum on the right-hand side converges. This space has the reproducing kernel

k_{w}(z) = ∑_{k=0}^{∞} \frac{(ν)_{k}}{k!} (zw̄)^{k} = \frac{1}{(1 − zw̄)^{ν}} ,

that is,

f(w) = ⟨f, k_{w}⟩, f ∈ A^{ν}, w ∈ 𝔻.

We denote the group action by

(f ∗ ϕ)(z) = f(ϕ(z))ϕ′(z)^{\frac{ν}{2}} = f (\frac{αz + β}{β̄z + ᾱ}) \frac{1}{(β̄z + ᾱ)^{ν}} ,

where ϕ ∈ G is given by

ϕ(z) = \frac{αz + β}{β̄z + ᾱ}, |α|^{2} − |β|^{2} = 1.

We also write

U_{ϕ}(f) = f ∗ ϕ^{−1}

for the corresponding left action. Because of the indeterminacy of the power function, one must pass to the universal covering group to get a unitary representation on each space A^{ν}. (If ν ∈ 2ℤ/k for an integer k, the k-fold cover is sufficient.)

It is easy to find joint eigenfunctions of our subgroups. Namely, since e^{iξz} are eigenfunctions of the translations, it follows that

e^{iξψ^{−1}(z)}((ψ^{−1})′(z))^{\frac{ν}{2}} ,

with ψ as above, are eigenfunctions of ϕ_{t}^{2}. It will be convenient to normalize them so that they take the value 1 at the origin. This gives the functions

e_{ξ}^{2}(z) = \frac{1}{(1 + z)^{ν}} exp (\frac{ξz}{1 + z} ) ,

which indeed satisfy

e_{ξ}^{2} ∗ ϕ_{t}^{2} = e^{itξ}e_{ξ}^{2}.

Replacing ψ by γ in this argument leads to the functions

e_{X}^{3}(z) = \frac{2^{\frac{ν}{2}}}{(1 − z)^{\frac{ν}{2} −iX}(1 + z)^{\frac{ν}{2} +iX}}

(the factor 2^{\frac{ν}{2}} being chosen for convenience), which satisfy

e_{X}^{3} ∗ ϕ_{t}^{3} = e^{itX}e_{X}^{3}.

Finally we write

e_{k}^{1}(z) = z^{k}

for the eigenfunctions of the elliptic subgroup. We call (e_{k}^{1})_{k=0}^{∞} the elliptic basis of A^{ν}, while (e_{ξ}^{2})_{ξ>0} and (e_{X}^{3})_{X∈ℝ} are called the parabolic and the hyperbolic (generalized) basis, respectively.

Note that, whereas clearly e_{k}^{1} is an element of A^{ν}, this is not true for e_{ξ}^{2} or e_{X}^{3}. However, one may still define ⟨f, e_{x}^{i}⟩ (i = 2, 3) for f in a dense subspace of A^{ν}. Natural choices are the space of polynomials and the linear hull of all reproducing kernels k_{z}, z ∈ D. With either of these choices the following Plancherel theorem is true.

Proposition 3.1. The operator (F_{2}f)(ξ) = ⟨f, e_{ξ}^{2}⟩ extends to a Hilbert space isomorphism

F_{2} : A^{ν} → L^{2} (ℝ_{+}, \frac{1}{Γ(ν)} e^{−ξ}ξ^{ν−1} dξ) .

Similarly, the operator (F_{3}f)(X) = ⟨f, e_{X}^{3}⟩ extends to an isomorphism

F_{3} : A^{ν} → L^{2} (ℝ, \frac{1}{2πΓ(ν)} |Γ (\frac{ν}{2} + iX)|^{2} dX) .

The transforms F_{2} and F_{3} are connected to the classical Fourier transform. For instance, F_{2}f is essentially the Fourier transform of the boundary value distribution of the function f(ψ(z))ψ′(z)^{ν/2} defined in the upper half-plane (they differ by a multiplicative factor). The proposition may be proved using this observation, cf. [P2], or alternatively using the orthogonality relations for Laguerre and Meixner–Pollaczek polynomials (whose role is made clear below).

By polarization, we have

⟨f, g⟩ = ∑_{k=0}^{∞} \frac{(ν)_{k}}{k!} ⟨f, e_{k}^{1}⟩⟨e_{k}^{1}, g⟩

(13)  = \frac{1}{Γ(ν)} ∫_{0}^{∞}⟨f, e_{ξ}^{2}⟩⟨e_{ξ}^{2}, g⟩ e^{−ξ}ξ^{ν−1} dξ
= \frac{1}{2πΓ(ν)} ∫_{−∞}^{∞}⟨f, e_{X}^{3}⟩⟨e_{X}^{3}, g⟩ |Γ (\frac{ν}{2} + iX)|^{2} dX,

with a suitable interpretation of the last two expressions. Choosing g = k_{z}, it follows that

f(z) = ∑_{k=0}^{∞} \frac{(ν)_{k}}{k!} (F_{1}f)(k)e_{k}^{1}(z)

(14)  = \frac{1}{Γ(ν)} ∫_{0}^{∞}(F_{2}f)(ξ)e_{ξ}^{2}(z)e^{−ξ}ξ^{ν−1} dξ
= \frac{1}{2πΓ(ν)} ∫_{−∞}^{∞}(F_{3}f)(X)e_{X}^{3}(z) |Γ (\frac{ν}{2} + iX)|^{2} dξ,

where

F_{1}f(k) = ⟨f, e_{k}^{1}⟩ = \frac{k!}{(ν)_{k}} f̂(k).

We now introduce the matrix elements

K^{ij}(x, y, ϕ) = ⟨e_{x}^{i}, e_{y}^{j} ∗ ϕ⟩, i, j = 1, 2, 3,

where ϕ ∈ G is such that this makes sense. By unitarity,

K^{ij}(x, y, ϕ) = \overline{K^{ji}(y, x, ϕ^{−1})},

so there are six cases to consider. As was remarked above, the K^{ij} were computed by Basu and Wolf [BW1]. We summarize their results in the following proposition. Proposition 3.2. The matrix elements of A^{ν} are given by

K^{11}(k, l, ϕ) = \begin{cases}\frac{β^{k}β̄^{l}}{α^{ν+k+l}} (−1)^{k} _{2}F_{1} ( \begin{matrix}−k, −l \\ ν \end{matrix} | − \frac{1}{|β|^{2}} ) & β ≠ 0, \\ \frac{k!}{(ν)_{k}α^{ν+2k}} δ_{kl} & β = 0, \end{cases}

K^{12}(k, ξ, ϕ) = \frac{(−1)^{k}(ᾱ + β)^{k}}{(α + β̄)^{ν+k}} exp (\frac{β̄ξ}{β̄ + α})_{1}F_{1} ( \begin{matrix}−k \\ ν \end{matrix} | \frac{ξ}{|α + β̄|^{2}} ) ,

K^{13}(k, X, ϕ) = \frac{2^{\frac{ν}{2}} (ᾱ − β)^{k}}{(α + β̄)^{\frac{ν}{2} −iX}(α − β̄)^{\frac{ν}{2} +iX+k}} _{2}F_{1} ( \begin{matrix}−k, \frac{ν}{2} − iX \\ ν \end{matrix} | \frac{2}{(α + β̄)(ᾱ − β)}) ,

K^{22}(ξ, η, ϕ) = \frac{1}{(2iIm(α − β))^{ν}} exp (−\frac{(ᾱ + β)ξ + (ᾱ − β̄)η}{2iIm(α − β)} )

×_{0}F_{1} ( \begin{matrix}− \\ ν \end{matrix} | − \frac{ξη}{4(Im(α − β))^{2}} ) ,

K^{23}(ξ, X, ϕ) = \frac{2^{\frac{ν}{2}}}{(2Re(α − β))^{\frac{ν}{2} +iX}(2iIm(α − β))^{\frac{ν}{2} −iX}} exp (\frac{(ᾱ − β)ξ}{2Re(α − β)})

×_{1}F_{1} ( \begin{matrix}\frac{ν}{2} − iX \\ ν \end{matrix} | \frac{iξ}{2Re(α − β)Im(α − β)}) ,

K^{33}(X, Y, ϕ) = \frac{(iIm(α + β))^{i(X−Y)}}{(Re(α − β))^{\frac{ν}{2} +iX}(Re(α + β))^{\frac{ν}{2} −iY}}

×_{2}F_{1} ( \begin{matrix}\frac{ν}{2} + iX, \frac{ν}{2} − iY \\ ν \end{matrix} | \frac{1}{Re(α + β)Re(α − β)}) ,

where ϕ(z) = (αz + β)/(β̄z + ᾱ), |α|^{2} − |β|^{2} = 1.

One may check that these expressions are well defined up to a choice of (ϕ′)^{ν/2}. The matrix elements may be identified with classical orthogonal functions as follows, cf. [K2]:

K^{11} : Meixner polynomials,
K^{12} : Laguerre polynomials,
K^{13} : Meixner–Pollaczek polynomials,
K^{22} : Bessel functions,
K^{23} : Laguerre functions,
K^{33} : Meixner–Pollaczek functions.

By inserting basis elements e_{x}^{i} ∗ ϕ into (13), one may deduce a host of addition theorems and orthogonality relations for these functions; cf. [VK], Section 7.7.

The proposition does not give the matrix elements K^{22}, K^{23}, or K^{33} for ϕ = id. However, by (14) it is natural to define

(15)  K^{22}(ξ, η, id) = Γ(ν)e^{ξ}ξ^{1−ν}δ(ξ − η),
(16)  K^{33}(X, Y, id) = 2πΓ(ν) |Γ (\frac{ν}{2} + iX)|^{−2} δ(X − Y),

where δ is the Dirac measure. Similarly one may define

K^{23}(ξ, X, id) = \frac{Γ(ν)e^{\frac{ξ}{2}}}{Γ(\frac{ν}{2} + iX)2^{iX}} ξ^{− \frac{ν}{2} +iX}

since this kernel has the property

e_{X}^{3}(z) = \frac{1}{Γ(ν)} ∫_{0}^{∞} \overline{K^{23}(ξ, X, id)} e_{ξ}^{2}(z)e^{−ξ}ξ^{ν−1} dξ.

(This reduces to the usual definition of the gamma function after a change of variables.) These identities can be made precise by extending the transforms F_{i} to suitable classes of distributions.

As for the proof, we remark that in the present realization, all of Proposition 3.2 may be deduced from Meixner’s expansion formula [Me]

(17) ∑_{k=0}^{∞} \frac{(ν)_{k}}{k!} _{2}F_{1} ( \begin{matrix}−k, α \\ ν \end{matrix} | x)_{2}F_{1} ( \begin{matrix}−k, β \\ ν \end{matrix} | y) z^{k}

= \frac{1}{(1 − z)^{ν−α−β}(1 − z + xz)^{α}(1 − z + yz)^{β}} _{2}F_{1} ( \begin{matrix}α, β \\ ν \end{matrix} | \frac{xyz}{(1 − z + xz)(1 − z + yz)}) .

The case x = 0 gives the expressions for K^{11} and K^{13}. To proceed, one may use the expansion

K^{33}(X, Y, ϕ) = ∑_{k=0}^{∞} \frac{(ν)_{k}}{k!} \overline{K^{13}(k, X, id)} K^{13}(k, Y, ϕ),

which follows from the first identity in (13). The expression for K^{33} given in Proposition 3.2 now follows from the general case of Meixner’s formula.

The matrix elements involving the parabolic subgroup are computed similarly, using two degenerate cases of Meixner’s formula. If we replace y by y/β in (17) and let β tend to infinity, we obtain the expansion

∑_{k=0}^{∞} \frac{(ν)_{k}}{k!} _{2}F_{1} ( \begin{matrix}−k, α \\ ν \end{matrix} | x)_{1}F_{1} ( \begin{matrix}−k \\ ν \end{matrix} | y) z^{k}

(18)

= \frac{1}{(1 − z)^{ν−α}(1 − z + xz)^{α}} exp (−\frac{yz}{1 − z} )_{1}F_{1} ( \begin{matrix}α \\ ν \end{matrix} | \frac{xyz}{(1 − z + xz)(1 − z)}) ,

due to Weisner. Repeating this once more gives the Hardy–Hille formula

∑_{k=0}^{∞} \frac{(ν)_{k}}{k!} _{1}F_{1} ( \begin{matrix}−k \\ ν \end{matrix} | x)_{1}F_{1} ( \begin{matrix}−k \\ ν \end{matrix} | y) z^{k}

= \frac{1}{(1 − z)^{ν}} exp (−\frac{z(x + y)}{1 − z})_{0}F_{1} ( \begin{matrix}− \\ ν \end{matrix} | \frac{xyz}{(1 − z)^{2}} ) .

We refer to the book [SM] for a large number of proofs of these formulas.

4. Coupling coefficients

In this section we will obtain the polynomials T_{i}Q as coupling coefficients of our representation. We consider transforms of the type

F_{i}^{ϕ}f(x) = F_{i} U_{ϕ}f(x) = ⟨f, e_{x}^{i} ∗ ϕ⟩, i = 1, 2, 3, ϕ ∈ G,

that is, the Fourier transform with respect to an arbitrary one-parameter subgroup of G. We will factorize these into operators which interact nicely with the Lie algebra action. More precisely, we will find factorizations of the type

F_{i}^{ϕ} = M_{i} T_{i} δ_{a}τ_{b},

where δ_{a} and τ_{b} are the dilatation and translation operators defined by

δ_{a}f(z) = f(az), τ_{b}f(z) = f(z + b),

M_{i} are the multiplication operators

M_{i}f(x) = \overline{e_{x}^{i} ∗ ϕ(0)}f(x) = (F_{i}^{ϕ}1)(x)f(x),

and T_{i} are the transforms defined in (4).

We will use the explicit expressions for the matrix elements given in Proposition 3.2 to obtain our factorizations. Note, however, that we only need them in the easiest special case, corresponding to x = 0 in (17) and (18).

Proposition 4.1. In the notation above, we have for ϕ = ( \begin{matrix}α & β \\ β̄ & ᾱ \end{matrix} ) the following factorizations of densely defined operators:

(19)  F_{1}^{ϕ} = M_{1} T_{1} δ_{1/αβ̄} τ_{−β/α} (β ≠ 0),
F_{2}^{ϕ} = M_{2} T_{2} δ_{1/(α+β̄)^{2}} τ_{−(ᾱ+β)/(α+β̄)},
F_{3}^{ϕ} = M_{3} T_{3} δ_{2/(α+β̄)(α−β̄)} τ_{(ᾱ−β)/(α−β̄)}.

Proof. Test the action of both sides on the function e_{k}^{1}(z) = z^{k}. In general,

(δ_{a}τ_{b} e_{k}^{1})(z) = (az + b)^{k} = b^{k} ∑_{j=0}^{k} \frac{(−k)_{j}}{j!} (−\frac{az}{b} )^{j} .

Thus one has for instance

M_{1} T_{1} δ_{1/αβ̄} τ_{−β/α} e_{k}^{1}(m) = \frac{β̄^{m}}{α^{ν+m}} T_{1} ((−\frac{β}{α})^{k} ∑_{j=0}^{k} \frac{(−k)_{j}}{j!} (\frac{z}{|β|^{2}} )^{j}) (m)

= \frac{β̄^{m}(−β)^{k}}{α^{ν+m+k}} ∑_{j=0}^{k} \frac{(−k)_{j}(−m)_{j}}{j!(ν)_{j}} (−\frac{1}{|β|^{2}} )^{j}

= \frac{β̄^{m}(−β)^{k}}{α^{ν+k+m}} _{2}F_{1} ( \begin{matrix}−k, −m \\ ν \end{matrix} | − \frac{1}{|β|^{2}} )
= K^{11}(k, m, ϕ) = F_{1}^{ϕ}e_{k}^{1}(m),

which proves (19). The other two identities follow similarly. □

Now let us apply these factorizations to a general tensor product A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}}. As in the case n = 1, we write

(f_{1} ⊗ ⋯ ⊗ f_{n}) ∗ ϕ(z) = f_{1}(ϕ(z_{1})) ⋯ f_{n}(ϕ(z_{n}))ϕ′(z_{1})^{\frac{ν_{1}}{2}} ⋯ ϕ′(z_{n})^{\frac{νn}{2}}

for the group action and F_{i}^{ϕ} for the transforms

(F_{i}^{ϕ}f)(x_{1}, …, x_{n}) = ⟨f, (e_{x_{1}}^{i} ⊗ ⋯ ⊗ e_{x_{n}}^{i}) ∗ ϕ⟩, i = 1, 2, 3, ϕ ∈ G.

Applying Proposition 4.1 in each variable gives factorizations of these transforms. We fix a highest weight polynomial Q of degree s and write

μ = |ν| + 2s.

Then there is a unique intertwining embedding K_{Q} of A^{μ} into A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} such that K_{Q}1 = Q. We will henceforth denote by ⟨ , ⟩_{μ} and ⟨ , ⟩_{ν} the scalar product of A^{μ} and A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}}, respectively.

Theorem 4.2. For g ∈ A^{μ} and ϕ ∈ G, we have the equalities

(F_{1}^{ϕ}K_{Q}g)(m) = T_{1}Q(m)(F_{1}^{ϕ}g)(|m| − s),
(F_{2}^{ϕ}K_{Q}g)(ξ) = T_{2}Q(ξ)(F_{2}^{ϕ}g)(|ξ|),
(F_{3}^{ϕ}K_{Q}g)(X) = T_{3}Q(X)(F_{3}^{ϕ}g)(|X|).

Thus we may write

⟨K_{Q}g, (z_{1}^{m_{1}} ⋯ z_{n}^{m_{n}}) ∗ ϕ⟩_{ν} = T_{1}Q(m_{1}, …, m_{n})⟨g, z^{|m|−s} ∗ ϕ⟩_{μ},

while for T_{2} and T_{3} similar equalities are valid with a suitable interpretation. In particular the matrix elements

⟨K_{Q}e_{x}^{i}, (e_{y_{1}}^{j} ⊗ ⋯ ⊗ e_{y_{n}}^{j}) ∗ ϕ⟩_{ν}

of the embedding K_{Q} factor as T_{j}Q(y) times a matrix element (in the sense of Section 3) for A^{|ν|+2s}.

Proof. Since K_{Q} commutes with the group action, we may assume that ϕ = id. Moreover, it suffices to choose g = k_{w} as a reproducing kernel. The first equality then reduces to

(20)  ⟨K_{Q}k_{w}, e_{m_{1}}^{1} ⊗ ⋯ ⊗ e_{m_{n}}^{1}⟩_{ν} = T_{1}Q(m)w̄^{|m|−s}.

First note that

k_{w} = (1 − |w|^{2})^{− \frac{μ}{2}} 1 ∗ ϕ_{w}^{−1},

where ϕ_{w} is the disc automorphism ϕ_{w}(z) = (z + w)/(1 + w̄z), corresponding to the matrix ( \begin{matrix}α & β \\ β̄ & ᾱ \end{matrix} ), with

α = \frac{1}{\sqrt{1 − |w|^{2}}} , β = \frac{w}{\sqrt{1 − |w|^{2}}} .

This gives

K_{Q}k_{w} = (1 − |w|^{2})^{− \frac{μ}{2}} Q ∗ ϕ_{w}^{−1}.

Thus we have

⟨K_{Q}k_{w}, e_{m_{1}}^{1} ⊗ ⋯ ⊗ e_{m_{n}}^{1}⟩_{ν} = (1 − |w|^{2})^{− \frac{μ}{2}} ⟨Q ∗ ϕ_{w}^{−1}, e_{m_{1}}^{1} ⊗ ⋯ ⊗ e_{m_{n}}^{1}⟩_{ν}
= (1 − |w|^{2})^{− \frac{μ}{2}} (F_{1}^{ϕ_{w}}Q)(m_{1}, …, m_{n}).

We now apply the factorization (19) to each variable. Note that (3) may be written as

(δ_{a}τ_{b})^{⊗n}Q = a^{s}Q.

This gives

⟨K_{Q}k_{w}, e_{m_{1}}^{1} ⊗ ⋯ ⊗ e_{m_{n}}^{1}⟩_{ν} = (1 − |w|^{2})^{− \frac{μ}{2}} \frac{β̄^{|m|}}{α^{|ν|+|m|}} (T_{1}(δ_{1/αβ̄} τ_{−β/α})^{⊗s}Q)(m)
= (1 − |w|^{2})^{− \frac{μ}{2}} \frac{β̄^{|m|−s}}{α^{|ν|+|m|+s}} T_{1}Q(m) = w̄^{|m|−s}T_{1}Q(m),

which proves (20). The rest of the theorem follows from the equalities

F_{2}^{ϕ_{w}}Q(ξ) = (1 − |w|^{2})^{\frac{μ}{2}} T_{2}Q(ξ) \overline{e_{|ξ|}^{2}(w)},
F_{3}^{ϕ_{w}}Q(X) = (1 − |w|^{2})^{\frac{μ}{2}} T_{3}Q(X) \overline{e_{|X|}^{3}(w)},

which are likewise easily obtained from Proposition 4.1. □

Choosing as in the proof g = k_{w}, ϕ = id, and expressing the left-hand sides by means of the adjoint K_{Q}^{∗}, one sees that the theorem is equivalent to the equalities

K_{Q}^{∗} e_{m_{1}}^{1} ⊗ ⋯ ⊗ e_{m_{n}}^{1} = \overline{T_{1}Q(m)} e_{|m|−s}^{1},
K_{Q}^{∗} e_{ξ_{1}}^{2} ⊗ ⋯ ⊗ e_{ξ_{n}}^{2} = \overline{T_{2}Q(ξ)} e_{|ξ|}^{2},
K_{Q}^{∗} e_{X_{1}}^{3} ⊗ ⋯ ⊗ e_{X_{n}}^{3} = \overline{T_{3}Q(X)} e_{|X|}^{3}.

This may also be proved using the explicit expression for K_{Q}^{∗} as a differential operator (transvectant) given in [R]; cf. [P2] for the case n = 2.

We now combine Theorem 4.2 with the equalities (14), choosing f = K_{Q}e_{x}^{i}. This will give the expansions of K_{Q}e_{x}^{i} in tensor products of the e_{y}^{j}. By the theorem, the coefficients will involve the polynomial T_{j}Q times the matrix element K^{ij}. For i = j this exhibits the polynomials T_{i}Q as coupling coefficients of our representation. We summarize this in a corollary.

Corollary 4.3. In the notation above, we have the generalized Clebsch–Gordan formulas

\frac{(μ)_{k}}{k!} K_{Q}e_{k}^{1}(z_{1}, …, z_{n}) = ∑_{|m|=k+s} \frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(m)z_{1}^{m_{1}} ⋯ z_{n}^{m_{n}},

\frac{ξ^{μ−1}}{Γ(μ)}K_{Q}e_{ξ}^{2}(z_{1}, …, z_{n})

(21)

= \frac{1}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{η∈ℝ_{+}^{n}: |η|=ξ} T_{2}Q(η) e_{η_{1}}^{2}(z_{1}) ⋯ e_{η_{n}}^{2}(z_{n})η_{1}^{ν_{1}−1} ⋯ η_{n}^{ν_{n}−1} dη,

\frac{|Γ (\frac{μ}{2} + iX)|^{2}}{Γ(μ)} K_{Q}e_{X}^{3}(z_{1}, …, z_{n}) = \frac{1}{(2π)^{n−1}Γ(ν_{1}) ⋯ Γ(ν_{n})}

× ∫_{Y ∈ℝ^{n}: |Y |=X} T_{3}Q(Y) e_{Y_{1}}^{3}(z_{1}) ⋯ e_{Y_{n}}^{3}(z_{n}) |Γ (\frac{ν_{1}}{2} + iY_{1}) ⋯ Γ (\frac{ν_{n}}{2} + iY_{n})|^{2} dY.

To obtain the latter two equalities, one may use formally (15) and (16). This can be justified either by a duality argument or by first writing down an analogous expansion for K_{Q}e_{x}^{i} ∗ ϕ and then letting ϕ tend to the identity.

Choosing i ≠ j gives six additional expansion formulas. These follow, however, from formulas which have been obtained already. For instance, i = 1, j = 2 gives

K_{Q}e_{k}^{1}(z) = \frac{(−1)^{k}}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{ℝ_{+}^{n}} _{1}F_{1} ( \begin{matrix}−k \\ μ \end{matrix} | |η|) T_{2}Q(η) e^{−|η|} ∏_{r=1}^{n} e_{η_{r}}^{2}(z_{r})η_{r}^{ν_{r}−1} dη.

Integrating over the hypersurfaces |η| = ξ and assuming (21), we see that this reduces to

K_{Q}e_{k}^{1}(z) = \frac{(−1)^{k}}{Γ(μ)} ∫_{0}^{∞} _{1}F_{1} ( \begin{matrix}−k \\ μ \end{matrix} | ξ) K_{Q}e_{ξ}^{2}(z)e^{−ξ}ξ^{μ−1} dξ.

However, this follows from applying K_{Q} to each side of

e_{k}^{1}(z) = \frac{1}{Γ(μ)} ∫_{0}^{∞} K^{12}(k, ξ, id) e_{ξ}^{2}(z)e^{−ξ}ξ^{μ−1} dξ,

which is a special case of (14).

As a consequence of the interpretation as coupling coefficients, we recover the orthogonality relations for the polynomials T_{i}Q.

Corollary 4.4. Let Q and Q̃ be two highest weight polynomials in

A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}},

not necessarily of the same degree. Then, if s is the degree of Q (or of Q̃) and μ = |ν| + 2s, the quantities

\frac{k!}{(μ)_{k}} ∑_{|m|=k+s} \frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(m) \overline{T_{1}Q̃(m)},

\frac{Γ(μ)B^{1−μ}}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{ξ∈ℝ_{+}^{n}: |ξ|=B} T_{2}Q(ξ) \overline{T_{2}Q̃(ξ)} ξ_{1}^{ν_{1}−1} ⋯ ξ_{n}^{ν_{n}−1} dξ,

\frac{Γ(μ)}{(2π)^{n−1} |Γ (\frac{μ}{2} + iC)|^{2}} ∫_{X∈ℝ^{n}: |X|=C} T_{3}Q(X) \overline{T_{3}Q̃(X)} ∏_{r=1}^{n} \frac{|Γ (\frac{ν_{r}}{2} + iX_{r})|^{2}}{Γ(ν_{r})} dX

are, for each k = 0, 1, 2, …, B ∈ ℝ_{+} and C ∈ ℝ, all equal to the scalar product ⟨Q, Q̃⟩_{ν}.

Proof. First note that, as a consequence of Schur’s lemma,

⟨K_{Q}f, KQ̃g⟩_{ν} = ⟨Q, Q̃⟩_{ν}⟨f, g⟩_{μ},

where we have used that K_{Q}1 = Q. If we put ξ = B in the equality (21) and then take the scalar product with KQ̃e_{B}^{2} ∗ ϕ for a suitable ϕ, the left-hand side thus equals

\frac{B^{μ−1}}{Γ(μ)} ⟨Q, Q̃⟩_{ν}⟨e_{B}^{2}, e_{B}^{2} ∗ ϕ⟩_{μ},

while the right-hand side equals

\frac{1}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{η∈ℝ_{+}^{n}: |η|=B} T_{2}Q(η)⟨e_{η_{1}}^{2} ⊗ ⋯ ⊗ e_{η_{n}}^{2}, KQ̃e_{B}^{2} ∗ ϕ⟩_{ν} η_{1}^{ν_{1}−1} ⋯ η_{n}^{ν_{n}−1} dη

= \frac{1}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{η∈ℝ_{+}^{n}: |η|=B} T_{2}Q(η) \overline{T_{2}Q̃(η)}⟨e_{B}^{2}, e_{B}^{2} ∗ ϕ⟩_{μ} η_{1}^{ν_{1}−1} ⋯ η_{n}^{ν_{n}−1} dη.

Dividing out the matrix element ⟨e_{B}^{2}, e_{B}^{2} ∗ ϕ⟩_{μ} we obtain the second part of the corollary. The remaining parts are similar. □

5. Convolution formulas

In this section we will obtain some convolution formulas for the polynomials T_{i}Q. The most general formulas of this type would follow from considering a scalar product

⟨K_{Q}e_{x}^{i} ∗ ϕ, (e_{y_{1}}^{j} ⊗ ⋯ ⊗ e_{y_{n}}^{j}) ∗ ψ⟩_{ν}

and expanding it by applying formula number k in (13) in each variable. For instance, with (i, j, k) = (1, 2, 3) this gives

⟨K_{Q}e_{k}^{1} ∗ ϕ, (e_{ξ_{1}}^{2} ⊗ ⋯ ⊗ e_{ξ_{n}}^{2}) ∗ ψ⟩_{ν}

= ∫_{ℝ^{n}}⟨K_{Q}e_{k}^{1} ∗ ϕ, e_{X_{1}}^{3} ⊗ ⋯ ⊗ e_{X_{n}}^{3}⟩_{ν}⟨e_{X_{1}}^{3} ⊗ ⋯ ⊗ e_{X_{n}}^{3}, (e_{ξ_{1}}^{2} ⊗ ⋯ ⊗ e_{ξ_{n}}^{2}) ∗ ψ⟩_{ν} dm(X),

where

dm(X) = \frac{1}{(2π)^{n}Γ(ν_{1}) ⋯ Γ(ν_{n})} |Γ (\frac{ν_{1}}{2} + iX_{1}) ⋯ Γ (\frac{ν_{n}}{2} + iX_{n})|^{2} dX_{1} ⋯ dX_{n}.

Now, by Theorem 4.2, this may be written as

T_{2}Q(ξ) K^{12}(k, |ξ|, ψϕ^{−1}) = ∫_{ℝ^{n}} T_{3}Q(X) K^{13}(k, |X|, ϕ^{−1}) ∏_{r=1}^{n} K^{32}(X_{r}, ξ_{r}, ψ) dm(X).

Inserting the expressions for matrix elements from Proposition 3.2, we obtain an integral formula linking T_{2}Q and T_{3}Q.

We will not write down all such formulas since they may be deduced from a few special cases. Using addition formulas for the matrix elements we may reduce ourselves to the case ϕ = id. As in the previous section, we may also assume that i = k.

In this case, the formulas follow from applying the transform F_{j}^{ψ} to both sides of the three generalized Clebsch–Gordan formulas of Corollary 4.3. Moreover, the formulas with j = 3 follow from those with j = 1, by means of (8). After these reductions, there remain the six cases

(i, j) = (1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (3, 2),

which will give the following convolution formulas. Theorem 5.1. Let, as above, Q be a highest weight polynomial in A^{ν_{1}} ⋯ ⊗ A^{ν_{n}} of degree s, and let μ = |ν| + 2s. Then the following formulas hold:

\frac{(μ)_{k}}{k!} c^{s}_{2}F_{1} ( \begin{matrix}−k, s − |l| \\ μ \end{matrix} | c) T_{1}Q(l) (22)

= ∑_{|m|=k+s} \frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(m) ∏_{r=1}^{n}_{2}F_{1} ( \begin{matrix}−m_{r}, −l_{r} \\ ν_{r} \end{matrix} | c) ,

\frac{(μ)_{k}}{k!} _{1}F_{1} ( \begin{matrix}−k \\ μ \end{matrix} | |ξ|) T_{2}Q(ξ)

= (−1)^{s} ∑_{|m|=k+s} \frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(m) ∏_{r=1}^{n}_{1}F_{1} ( \begin{matrix}−m_{r} \\ ν_{r} \end{matrix} | ξ_{r}) ,

\frac{ξ^{μ−1}}{Γ(μ)} _{1}F_{1} ( \begin{matrix}s − |m| \\ μ \end{matrix} | ξ) T_{1}Q(m) (23)

= \frac{(−1)^{s}}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{η∈ℝ_{+}^{n}: |η|=ξ} T_{2}Q(η) ∏_{r=1}^{n}_{1}F_{1} ( \begin{matrix}−m_{r} \\ ν_{r} \end{matrix} | η_{r}) η_{1}^{ν_{1}−1} ⋯ η_{n}^{ν_{n}−1} dη,

\frac{ξ^{μ−1}}{Γ(μ)} _{0}F_{1} ( \begin{matrix}− \\ μ \end{matrix} | − ξ|ζ|) T_{2}Q(ζ)

= \frac{(−1)^{s}}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{η∈ℝ_{+}^{n}: |η|=ξ} T_{2}Q(η) ∏_{r=1}^{n}_{0}F_{1} ( \begin{matrix}− \\ ν_{r} \end{matrix} | − η_{r}ζ_{r}) η_{1}^{ν_{1}−1} ⋯ η_{n}^{ν_{n}−1} dη,

\frac{|Γ (\frac{μ}{2} + iX)|^{2}}{Γ(μ)} c^{s}_{2}F_{1} ( \begin{matrix}s − |m|, \frac{μ}{2} + iX \\ μ \end{matrix} | c) T_{1}Q(m) (24)

= \frac{1}{(2π)^{n−1}} ∫_{Y ∈ℝ^{n}: |Y |=X} T_{3}Q(Y) ∏_{r=1}^{n} \frac{|Γ (\frac{ν_{r}}{2} + iY_{r})|^{2}}{Γ(ν_{r})} _{2}F_{1} ( \begin{matrix}−m_{r}, \frac{ν_{r}}{2} + iY_{r} \\ ν_{r} \end{matrix} | c) dY,

\frac{|Γ (\frac{μ}{2} + iX)|^{2}}{Γ(μ)} _{1}F_{1} ( \begin{matrix}\frac{μ}{2} + iX \\ μ \end{matrix} | |ξ|) T_{2}Q(ξ)

= \frac{i^{s}}{(2π)^{n−1}} ∫_{Y ∈ℝ^{n}: |Y |=X} T_{3}Q(Y) ∏_{r=1}^{n} \frac{|Γ (\frac{ν_{r}}{2} + iY_{r})|^{2}}{Γ(ν_{r})} _{1}F_{1} ( \begin{matrix}\frac{ν_{r}}{2} + iY_{r} \\ ν_{r} \end{matrix} | iξ_{r}) dY.

The parameter c occurring in (22) and (24) depends on the choice of ψ. More precisely, in the first of these identities c = −1/|β|^{2}, where ψ(z) = (αz +β)/(β̄z +ᾱ), and in the second one c = 2/(α + β̄)(ᾱ − β). Since both sides are polynomials in c they are valid for arbitrary c. For the remaining four identities, different choices of ψ give essentially the same result.

Quite a lot of information is contained in these formulas. Many known identities follow from the three interesting special cases when (i) Q = 1, (ii) n = 2, or (iii) a numerator parameter in the hypergeometric function on the left-hand side is 0, so that this function reduces to 1. In particular, the case n = 2 of the first two identities occur in [KV1] with a similar group theoretic interpretation; cf. also [Su], [VdJ].

It should be noted that if Q is constructed by binary coupling as in Subsection 2.4, the theorem may be proved using the case n = 2 and induction on n. Since one may construct a basis for the space of highest weight vectors in this way, this actually gives an alternative proof of the general case.

Some degenerate cases of our convolution formulas seem interesting enough to state explicitly. First, the polynomials T_{1}Q (and thus T_{3}Q) satisfy some related identities.

Corollary 5.2. In the notation of Theorem 5.1, the following identities are valid:

(25) \frac{(μ + |l| − s)_{k}}{k!} T_{1}Q(l) = ∑_{|m|=k+s} \frac{(ν_{1} + l_{1})_{m_{1}} ⋯ (ν_{n} + l_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(m),

(26) \frac{(μ + |l| − s)_{k+s}}{(k + s)!} T_{1}Q(l) = ∑_{|m|=k+s} \frac{(ν_{1} + l_{1})_{m_{1}} ⋯ (ν_{n} + l_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(m + l),

(27)  \frac{(s − |l|)_{k}}{k!} T_{1}Q(l) = (−1)^{s} ∑_{|m|=k+s} \frac{(−l_{1})_{m_{1}} ⋯ (−l_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(m),

(28)  \frac{(s − |l|)_{k+s}}{(k + s)!} T_{1}Q(l) = ∑_{|m|=k+s} \frac{(−l_{1})_{m_{1}} ⋯ (−l_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(l − m).

Proof. This follows from the convolution formula (22). As was remarked above, it is valid for arbitrary values of c, though in the proof c = −1/|β|^{2} is negative. Putting c = 1 and using the Chu–Vandermonde formula

2F_{1} ( \begin{matrix}−m, −l \\ ν \end{matrix} | 1) = \frac{(ν)_{m+l}}{(ν)_{m}(ν)_{l}}

gives the equality (25). If we multiply both sides of (22) by c^{−k−s} and use the limit relation

lim_{c→∞} \frac{1}{c^{m}} _{2}F_{1} ( \begin{matrix}−m, −l \\ ν \end{matrix} | c) = (−1)^{m} \frac{(−l)_{m}}{(ν)_{m}} (m, l = 0, 1, 2, …),

we obtain (27). Incidentally, (27) also follows from (25) together with

(29)  T_{1}Q(m) = (−1)^{s} T_{1}Q(−ν − m),

which in turn follows from (8) and (9).

If we instead multiply both sides of (22) by c^{−|l|} and let c → ∞, we get

(−1)^{s} \frac{(μ)_{k}(−k)_{|l|−s}}{k!(μ)_{|l|−s}} T_{1}Q(l) = ∑_{|m|=k+s} \frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}(−m_{1})_{l_{1}} ⋯ (−m_{n})_{l_{n}}}{m_{1}! ⋯ m_{n}! (ν_{1})_{l_{1}} ⋯ (ν_{n})_{l_{n}}} T_{1}Q(m).

Replacing k by k + |l| and m by m + l gives (26). The equality (28) follows from (26) together with (29). □

We may also obtain two integral formulas for T_{1}Q from Theorem 5.1. In the bilinear case, the first of these is a known formula for the Hahn polynomials, cf. [VK], equation 8.2.4(7), while we have not found the second one in the literature.

Corollary 5.3. In the notation of Theorem 5.1, the following identities are valid:

\frac{1}{Γ(μ + |m| − s)} T_{1}Q(m)

= \frac{1}{Γ(ν_{1} + m_{1}) ⋯ Γ(ν_{n} + m_{n})} ∫_{η∈ℝ_{+}^{n}: |η|=1} T_{2}Q(η) η_{1}^{m_{1}+ν_{1}−1} ⋯ η_{n}^{m_{n}+ν_{n}−1} dη,

\frac{Γ (\frac{μ}{2} + iX) Γ (\frac{μ}{2} − iX + |m| − s)}{Γ(μ + |m| − s)} T_{1}Q(m)

= \frac{1}{(2π)^{n−1}} ∫_{Y ∈ℝ^{n}: |Y |=X} T_{3}Q(Y) ∏_{r=1}^{m} \frac{Γ (\frac{ν_{r}}{2} + iY_{r}) Γ (\frac{ν_{r}}{2} − iY_{r} + m_{r})}{Γ(ν_{r} + m_{r})} dY.

Proof. This follows from the convolution formulas (23) and (24), respectively. In the integral in (23), replace η by ξη so that the integration is over {|η| = 1}. Then let ξ → ∞ and use the homogeneity of T_{2}Q and the limit relation

lim_{ξ→∞} \frac{1}{ξ^{m}} _{1}F_{1} ( \begin{matrix}−m \\ ν \end{matrix} | ξη) = \frac{(−1)^{m}}{(ν)_{m}} η^{m} (m = 0, 1, 2, …).

The other half of the corollary is the case c = 1 of (24). □

6. Coupling kernels

In this section we introduce some polynomials in 2n variables which we call coupling kernels. Let V_{s} be the subspace of A^{ν_{1}} ⊗⋯⊗A^{ν_{n}} consisting of highest weight vectors of degree s, and let Q_{s}(z, w̄) be its reproducing kernel. The coupling kernels are obtained by applying one of the transforms T_{i} to each variable in Q_{s}(z, w̄). If

Q_{s}(z, w̄) = ∑_{|t|=|u|=s} c_{t,u} w̄^{t}z^{u},

we define the coupling kernel P_{s}^{ij} by

P_{s}^{ij}(x, y) = ∑_{|t|=|u|=s} c_{t,u} \overline{(T_{i}w^{t})(x)} (T_{j}z^{u})(y), i, j = 1, 2, 3.

Recall the decomposition

A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} ≃ ⨁_{s=0}^{∞} \binom{n + s − 2}{n − 2}A^{|ν|+2s}.

Let Π_{s} denote the orthogonal projection of A^{ν_{1}} ⊗⋯⊗A^{ν_{n}} onto the isotypic subspace equivalent to \binom{n+s−2}{n−2}A^{|ν|+2s}. By general Hilbert space arguments, we may express the reproducing kernel Q_{s} as

Q_{s}(z, w̄) = ∑_{|t|=|u|=s} \frac{(ν)_{t}(ν)_{u}}{t! u!} ⟨Π_{s}z^{t}, z^{u}⟩_{ν} w̄^{t}z^{u}

or as

Q_{s}(z, w̄) = ∑_{k∈Λ_{s}} \overline{Q_{k}(w)} Q′_{k}(z),

where (Q_{k})_{k∈Λ_{s}} is any basis of V_{s} and (Q′_{k})_{k∈Λ_{s}} is the dual basis (here Λ_{s} is a suitable index set of cardinality \binom{n+s−2}{n−2}). Thus the coupling kernels are given by

(30)  P_{s}^{ij}(x, y) = ∑_{|t|=|u|=s} \frac{(ν)_{t}(ν)_{u}}{t! u!} ⟨Π_{s}z^{t}, z^{u}⟩_{ν}\overline{(T_{i}w^{t})(x)} (T_{j}z^{u})(y)
(31)  = ∑_{k∈Λ_{s}} \overline{T_{i}Q_{k}(x)} T_{j}Q′_{k}(y).

It will turn out that the coupling kernels may be expressed more explicitly in terms of Karlsson’s generalized Kampé de Fériet functions (10). Moreover, we will recover as very special cases the dual Appell polynomials described in Subsection 2.3.

To motivate the introduction of the coupling kernels, consider the matrix elements of the projection Π_{s}:

(32)  ⟨Π_{s}(e_{x_{1}}^{i} ⊗ ⋯ ⊗ e_{x_{n}}^{i}), (e_{y_{1}}^{j} ⊗ ⋯ ⊗ e_{y_{n}}^{j}) ∗ ϕ⟩_{ν}, i, j = 1, 2, 3.

One may express Π_{s} as

Π_{s} = ∑_{k∈Λ_{s}} K_{Q′_{k}}K_{Q_{k}}^{∗},

where (Q_{k}) and (Q′_{k}) are as above. Using Theorem 4.2 one finds that the matrix element (32) equals a matrix element (in the sense of Section 3) for A^{|ν|+2s} times the coupling kernel P_{s}^{ij}(x, y). For instance, one has

⟨Π_{s}(e_{l_{1}}^{1} ⊗ ⋯ ⊗ e_{l_{n}}^{1}), (e_{ξ_{1}}^{2} ⊗ ⋯ ⊗ e_{ξ_{n}}^{2}) ∗ ϕ⟩_{ν}
= ∑_{k∈Λ_{s}}⟨K_{Q_{k}}^{∗}e_{l_{1}}^{1} ⊗ ⋯ ⊗ e_{l_{n}}^{1}, K_{Q′_{k}}^{∗}(e_{ξ_{1}}^{2} ⊗ ⋯ ⊗ e_{ξ_{n}}^{2}) ∗ ϕ⟩_{|ν|+2s}
= ∑_{k∈Λ_{s}} \overline{T_{1}Q_{k}(l)} T_{2}Q′_{k}(ξ)⟨e_{|l|−s}^{1}, e_{|ξ|}^{2} ∗ ϕ⟩_{|ν|+2s}
= K^{12}(|l| − s, |ξ|, ϕ) P_{s}^{12}(l, ξ).

We will now give explicit expressions for the coupling kernels. It is clear that

(33)  P_{s}^{ij}(x, y) = \overline{P_{s}^{ji}(y, x)},

so we may assume that i ≤ j. Moreover, it follows from (8) that

(34)  P_{s}^{i3}(x, Y) = P_{s}^{i1} (x, iY − \frac{ν}{2}) ,

which leaves us with the three cases P_{s}^{11}, P_{s}^{12}, and P_{s}^{22}.

Theorem 6.1. The coupling kernels are given in terms of Karlsson’s functions (10) by

P_{s}^{11}(k, l) = \frac{(−1)^{s}(−|k|)_{s}(−|l|)_{s}}{(|ν| + s − 1)_{s}s!} F_{2:1}^{2:2} ( \begin{matrix}|ν| + s − 1, −s & : & −k, −l \\ −|k|, −|l| & : & ν \end{matrix} | 1) ,
P_{s}^{12}(k, ξ) = \frac{(−|k|)_{s}|ξ|^{s}}{(|ν| + s − 1)_{s}s!} F_{1:1}^{2:1} ( \begin{matrix}|ν| + s − 1, −s & : & −k \\ −|k| & : & ν \end{matrix} | \frac{ξ}{|ξ|}) ,
P_{s}^{22}(ξ, η) = \frac{(−1)^{s}|ξ|^{s}|η|^{s}}{(|ν| + s − 1)_{s}s!} F_{0:1}^{2:0} ( \begin{matrix}|ν| + s − 1, −s & : & − \\ − & : & ν \end{matrix} | \frac{ξη}{|ξ||η|}) .

The proof of this theorem will be deferred to Section 8. The variable of the F_{0:1}^{2:0}- function should be interpreted as

\frac{ξη}{|ξ||η|} = \frac{1}{(ξ_{1} + ⋯ + ξ_{n})(η_{1} + ⋯ + η_{n})} (ξ_{1}η_{1}, …, ξ_{n}η_{n}) .

We also remark that Karlsson’s F_{0:1}^{2:0}-function is Lauricella’s F_{C}-function.

These expressions should be compared with what one gets from choosing a particular basis (Q_{k}) in (31). This is particularly interesting when n = 2, when the sum has only one term. In the case of P_{s}^{22} one gets an expression for the product of two Jacobi polynomials as an F_{0:1}^{2:0}-series of two variables, that is, as an Appell F_{4}-series. This is a classical result of Watson [W]. For P_{s}^{11} one gets a similar expression for the product of two Hahn polynomials due to Gasper [G1]; cf. also [G2]. For P_{s}^{12} one gets the identity

\frac{(−1)^{s}s!}{(ν_{2})_{s}} P_{s}^{(ν_{1}−1,ν_{2}−1)} (\frac{ξ_{2} − ξ_{1}}{ξ_{1} + ξ_{2}} ) Q_{s}(k_{1}; ν_{1} − 1, ν_{2} − 1; k_{1} + k_{2})

= F_{1:1}^{2:1} ( \begin{matrix}ν_{1} + ν_{2} + s − 1, −s & : & −(k_{1}, k_{2}) \\ −k_{1} − k_{2} & : & (ν_{1}, ν_{2}) \end{matrix} | \frac{ξ_{1}}{ξ_{1} + ξ_{2}} , \frac{ξ_{2}}{ξ_{1} + ξ_{2}} )

or equivalently

2F_{1} ( \begin{matrix}−n, n + a \\ b \end{matrix} | z)_{3}F_{2} ( \begin{matrix}−n, n + a, c \\ b, d \end{matrix} | 1)

= \frac{(−1)^{n}(a − b + 1)_{n}}{(b)_{n}} F_{1:1}^{2:1} ( \begin{matrix}−n, n + a & : & (c, d − c) \\ d & : & (b, a − b + 1) \end{matrix} | z, 1 − z) ,

which we have not found in the literature. For general n one may obtain generalizations of these formulas by choosing in (31) the basis (11) of V_{s} or a basis constructed by binary coupling as in Subsection 2.4.

As was mentioned above, the dual Appell polynomials described in Subsection 2.3 may be viewed as special cases of coupling kernels. This might first seem surprising, but is easily understood.

Corollary 6.2. Let (Q_{t}) be the basis

Q_{t}(z) = (z_{1} − z_{n})^{t_{1}} ⋯ (z_{n−1} − z_{n})^{t_{n−1}}, |t| = t_{1} + ⋯ + t_{n−1} = s

of the space of highest weight polynomials in A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} of degree s. Let (Q′_{t}) be the dual basis. For x = (x_{1}, …, x_{n}), let x̂ = (x_{1}, …, x_{n−1}). Then the polynomials T_{i}Q′_{t} may be expressed in terms of Karlsson’s functions as

T_{1}Q′_{t}(m) = \frac{(ν̂)_{t}(−|m|)_{s}}{t! (|ν| + s − 1)_{s}} F_{1:1}^{1:2} ( \begin{matrix}|ν| + s − 1 & : & −t, −m̂ \\ −|m| & : & ν̂ \end{matrix} | 1) ,
T_{2}Q′_{t}(ξ) = \frac{(ν̂)_{t}(−|ξ|)^{s}}{t! (|ν| + s − 1)_{s}} F_{0:1}^{1:1} ( \begin{matrix}|ν| + s − 1 & : & −t \\ − & : & ν̂ \end{matrix} | \frac{ξ_{1}}{|ξ|}, …, \frac{ξ_{n−1}}{|ξ|} ) ,

T_{3}Q′_{t}(X) = \frac{(ν̂)_{t}(\frac{|ν|}{2} − i|X|)_{s}}{t! (|ν| + s − 1)_{s}} F_{1:1}^{1:2} ( \begin{matrix}|ν| + s − 1 & : & −t, \frac{1}{2} ν̂ − iX̂ \\ \frac{1}{2}|ν| − i|X| & : & ν̂ \end{matrix} | 1) .

Note that the vector parameters in Karlsson’s functions have dimension n − 1 here but dimension n in Theorem 6.1.

Proof. If u = (u_{1}, …, u_{n}), |u| = s, we may write

Π_{s}z^{u} = ∑_{v}⟨z^{u}, Q_{v}⟩_{ν} Q′_{v} = ∑_{v} \overline{T_{1}Q_{v}(u)} Q′_{v}.

Thus

(35)  T_{i}Π_{s}z^{u}(x) = ∑_{v} \overline{T_{1}Q_{v}(u)} T_{i}Q′_{v}(x) = P_{s}^{1i}(u, x).

Now choose

(36)  u = (t, 0) = (t_{1}, …, t_{n−1}, 0), |t| = s.

Then

⟨z^{u}, Q_{v}⟩_{ν} = \frac{t_{1}! ⋯ t_{n−1}!}{(ν_{1})_{t_{1}} ⋯ (ν_{n−1})_{t_{n−1}}} δ_{t,v},

so that in fact

(37)  Q′_{t} = \frac{(ν_{1})_{t_{1}} ⋯ (ν_{n−1})_{t_{n−1}}}{t_{1}! ⋯ t_{n−1}!} Π_{s}z^{u}.

Combining (35) and (37) gives

T_{i}Q′_{t}(x) = \frac{(ν_{1})_{t_{1}} ⋯ (ν_{n−1})_{t_{n−1}}}{t_{1}! ⋯ t_{n−1}!} P_{s}^{1i}(u, x), i = 1, 2, 3.

It is now easy to check that the expressions for P_{s}^{1i}(u, x) given in Theorem 6.1 simplify to those of the corollary when u is of the form (36). □

7. Further properties of the coupling kernels

We continue to study the formal properties of the coupling kernels. Some formulas follow from Corollary 4.4 and Theorem 5.1 by choosing highest weight vectors Q of the form

Q_{isx} = ∑_{k∈Λ_{s}} \overline{T_{i}Q_{k}(x)} Q′_{k},

where, as above, (Q_{k})_{k∈Λ_{s}} is any basis of V_{s} and (Q′_{k})_{k∈Λ_{s}} the dual basis. For such Q one has

⟨Q, Q_{isx}⟩_{ν} = T_{i}Q(x), Q ∈ V_{s},

so Q_{isx} is the kernel of T_{i}, viewed as an operator on V_{s}. We will write down these formulas explicitly since we find them quite interesting.

Applying Corollary 4.4 with Q̃ = Q_{isx} gives the following identities.

Proposition 7.1. Let Q be a highest weight polynomial in A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} of degree s, and let μ = |ν| + 2s. Then, for i = 1, 2, 3, k = 0, 1, 2, …, B ∈ ℝ_{+}, and C ∈ ℝ, one has the equalities

T_{i}Q(x) = \frac{k!}{(μ)_{k}} ∑_{|m|=k+s} \frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T_{1}Q(m) P_{s}^{1i}(x, m)

= \frac{Γ(μ)B^{1−μ}}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{ξ∈ℝ_{+}^{n}: |ξ|=B} T_{2}Q(ξ) P_{s}^{2i}(x, ξ) ξ_{1}^{ν_{1}−1} ⋯ ξ_{n}^{ν_{n}−1} dξ

= \frac{Γ(μ)}{(2π)^{n−1} |Γ (\frac{μ}{2} + iC)|^{2}} ∫_{X∈ℝ^{n}: |X|=C} T_{3}Q(X) P_{s}^{3i}(x, X) ∏_{r=1}^{n} \frac{|Γ (\frac{ν_{r}}{2} + iX_{r})|^{2}}{Γ(ν_{r})} dX.

In particular, P_{s}^{ii} is essentially the reproducing kernel for the space of coupling coefficients T_{i}Q of degree s. If we recall the connection between the polynomials T_{2}Q and spherical harmonics, we see that the reproducing kernels for Vilenkin’s polyspherical harmonics may be expressed in terms of P_{s}^{22} and thus in terms of Karlsson’s function F_{0:1}^{2:0}. This is an interesting fact which we have not found in the literature.

If we choose Q̃ = Q_{isx} and Q = Q_{jty} in Corollary 4.4, we obtain addition formulas for the coupling kernels.

Proposition 7.2. In the notation above, one has for i, j = 1, 2, 3 the equalities

P_{s}^{ij}(x, y) δ_{s,t} = \frac{k!}{(μ)_{k}} ∑_{|m|=k+s} \frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} P_{s}^{i1}(x, m)P_{t}^{1j}(m, y)

= \frac{Γ(μ)B^{1−μ}}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{ξ∈ℝ_{+}^{n}: |ξ|=B} P_{s}^{i2}(x, ξ)P_{t}^{2j}(ξ, y) ξ_{1}^{ν_{1}−1} ⋯ ξ_{n}^{ν_{n}−1} dξ

= \frac{Γ(μ)}{(2π)^{n−1} |Γ (\frac{μ}{2} + iC)|^{2}} ∫_{X∈ℝ^{n}: |X|=C} P_{s}^{i3}(x, X)P_{t}^{3j}(X, y) ∏_{r=1}^{n} \frac{|Γ (\frac{ν_{r}}{2} + iX_{r})|^{2}}{Γ(ν_{r})} dX.

If we choose Q = Q_{isx} in Theorem 5.1, we obtain eighteen convolution formulas.

Proposition 7.3. In the notation above, one has for i = 1, 2, 3 the equalities

\frac{(μ)_{k}}{k!} c^{s}_{2}F_{1} ( \begin{matrix}−k, s − |l| \\ μ \end{matrix} | c) P_{s}^{i1}(x, l)

= ∑_{|m|=k+s} \frac{(ν)_{m}}{m!} P_{s}^{i1}(x, m) ∏_{r=1}^{n}_{2}F_{1} ( \begin{matrix}−m_{r}, −l_{r} \\ ν_{r} \end{matrix} | c) ,

\frac{(μ)_{k}}{k!} _{1}F_{1} ( \begin{matrix}−k \\ μ \end{matrix} | |ξ|) P_{s}^{i2}(x, ξ)

= (−1)^{s} ∑_{|m|=k+s} \frac{(ν)_{m}}{m!} P_{s}^{i1}(x, m) ∏_{r=1}^{n}_{1}F_{1} ( \begin{matrix}−m_{r} \\ ν_{r} \end{matrix} | ξ_{r}) ,

\frac{ξ^{μ−1}}{Γ(μ)} _{1}F_{1} ( \begin{matrix}s − |m| \\ μ \end{matrix} | |ξ|) P_{s}^{i1}(x, m)

= \frac{(−1)^{s}}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{η∈ℝ_{+}^{n}: |η|=ξ} P_{s}^{i2}(x, η) ∏_{r=1}^{n}_{1}F_{1} ( \begin{matrix}−m_{r} \\ ν_{r} \end{matrix} | η_{r}) η_{1}^{ν_{1}−1} ⋯ η_{n}^{ν_{n}−1} dη,

\frac{ξ^{μ−1}}{Γ(μ)} _{0}F_{1} ( \begin{matrix}− \\ μ \end{matrix} | − ξ|ζ|) P_{s}^{i2}(x, ζ)

= \frac{(−1)^{s}}{Γ(ν_{1}) ⋯ Γ(ν_{n})} ∫_{η∈ℝ_{+}^{n}: |η|=ξ} P_{s}^{i2}(x, η) ∏_{r=1}^{n}_{0}F_{1} ( \begin{matrix}− \\ ν_{r} \end{matrix} | − η_{r}ζ_{r}) η_{1}^{ν_{1}−1} ⋯ η_{n}^{ν_{n}−1} dη,

\frac{|Γ (\frac{μ}{2} + iX)|^{2}}{Γ(μ)} c^{s}_{2}F_{1} ( \begin{matrix}s − |m|, \frac{μ}{2} + iX \\ μ \end{matrix} | c) P_{s}^{i1}(x, m)

= \frac{1}{(2π)^{n−1}} ∫_{Y ∈ℝ^{n}: |Y |=X} P_{s}^{i3}(x, Y) ∏_{r=1}^{n} \frac{|Γ (\frac{ν_{r}}{2} + iY_{r})|^{2}}{Γ(ν_{r})} _{2}F_{1} ( \begin{matrix}−m_{r}, \frac{ν_{r}}{2} + iY_{r} \\ ν_{r} \end{matrix} | c) dY,

\frac{|Γ (\frac{μ}{2} + iX)|^{2}}{Γ(μ)} _{1}F_{1} ( \begin{matrix}\frac{μ}{2} + iX \\ μ \end{matrix} | i|ξ|) P_{s}^{i2}(x, ξ)

= \frac{i^{s}}{(2π)^{n−1}} ∫_{Y ∈ℝ^{n}: |Y |=X} P_{s}^{i3}(x, Y) ∏_{r=1}^{n} \frac{|Γ (\frac{ν_{r}}{2} + iY_{r})|^{2}}{Γ(ν_{r})} _{1}F_{1} ( \begin{matrix}\frac{ν_{r}}{2} + iY_{r} \\ ν_{r} \end{matrix} | iξ_{r}) dY.

Degenerate cases of these formulas are obtained by choosing Q = Q_{isx} in Corollaries 5.2 and 5.3.

We will now give some expansions which are different in nature from those obtained so far. They are of a type known as linearization formulas. Let Π_{s} be the orthogonal projection introduced in Section 6. Then the scalar product of A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} decomposes as

(38)  ⟨f, g⟩_{ν} = ∑_{s=0}^{∞}⟨Π_{s}f, g⟩_{ν}.

We insert in this equality f = e_{x_{1}}^{i} ⊗ ⋯ ⊗ e_{x_{n}}^{i}, g = (e_{y_{1}}^{i} ⊗ ⋯ ⊗ e_{y_{n}}^{i}) ∗ ϕ. Then the left-hand side is a product of matrix elements for the spaces A^{ν_{i}}, while, as we have observed, each term on the right-hand side is the product of a coupling kernel P_{s}^{ij}(x, y) and a matrix element for A^{|ν|+2s}. In view of (33) and (34) it suffices to consider the three cases

(i, j) = (1, 1), (1, 2), (2, 2).

This gives the following linearization formulas.

Proposition 7.4. In the notation above,

(39) ∏_{r=1}^{n}_{2}F_{1} ( \begin{matrix}−k_{r}, −l_{r} \\ ν_{r} \end{matrix} | c) = ∑_{s=0}^{min(|k|,|l|)} P_{s}^{11}(k, l) c^{s}_{2}F_{1} ( \begin{matrix}s − |k|, s − |l| \\ |ν| + 2s \end{matrix} | c) ,

∏_{r=1}^{n}_{1}F_{1} ( \begin{matrix}−k_{r} \\ ν_{r} \end{matrix} | ξ_{r}) = ∑_{s=0}^{|k|} P_{s}^{12}(k, ξ)(−1)^{s}_{1}F_{1} ( \begin{matrix}s − |k| \\ |ν| + 2s \end{matrix} | |ξ|) ,

∏_{r=1}^{n}_{0}F_{1} ( \begin{matrix}− \\ ν_{r} \end{matrix} | − ξ_{r}η_{r}) = ∑_{s=0}^{∞} P_{s}^{22}(ξ, η)_{0}F_{1} ( \begin{matrix}− \\ |ν| + 2s \end{matrix} | − |ξ||η|) .

For n = 2, when the coupling kernels factor as a product of two Clebsch–Gordan coefficients, the first of these formulas is the Burchnall–Chaundy formula [BC], while the third one is attributed to Bateman in [VK]. The bilinear case is also treated in [KV2].

It is not hard to prove these formulas directly, or, in fact, to obtain them as special cases of more general expansion formulas. For instance, one may prove that (assuming convergence)

∏_{r=1}^{n}_{2}F_{1} ( \begin{matrix}A_{r}, B_{r} \\ C_{r} \end{matrix} | x)

(40)

= ∑_{s=0}^{∞} \frac{(a)_{s}(b)_{s}(−1)^{s}}{s!(c + s − 1)_{s}} F_{2:1}^{2:2} ( \begin{matrix}c + s − 1, −s & : & A, B \\ a, b & : & C \end{matrix} | 1) x^{s}_{2}F_{1} ( \begin{matrix}a + s, b + s \\ c + 2s \end{matrix} | x) ,

which reduces to (39) for special values of the parameters and to Chaundy’s formula [C]

2F_{1} ( \begin{matrix}A, B \\ C \end{matrix} | x)

(41)

= ∑_{s=0}^{∞} \frac{(a)_{s}(b)_{s}(−1)^{s}}{s!(c + s − 1)_{s}} _{4}F_{3} ( \begin{matrix}c + s − 1, −s, A, B \\ a, b, C \end{matrix} | 1) x^{s}_{2}F_{1} ( \begin{matrix}a + s, b + s \\ c + 2s \end{matrix} | x)

for n = 1. In fact, (40) is a special case of an even more general expansion formula due to H. M. Srivastava [S]; cf. also [SK], pp. 340–341. (I would like to thank Professor Srivastava for pointing this out to me.) In [R], we gave a group theoretic interpretation of (41), where the _{4}F_{3}-series occur as Racah coefficients (or Wigner 6-j-symbols). It would be interesting to have an interpretation of the more general formula (40).

We finally note some degenerate cases of (39). If we put c = 1, we get

\frac{(ν_{1})_{k_{1}+l_{1}} ⋯ (ν_{n})_{k_{n}+l_{n}}}{(ν_{1})_{k_{1}} ⋯ (ν_{n})_{k_{n}}(ν_{1})_{l_{1}} ⋯ (ν_{n})_{l_{n}}} = ∑_{s=0}^{min(|k|,|l|)} \frac{(|ν| + 2s)_{|k|+|l|−2s}}{(|ν| + 2s)_{|k|−s}(|ν| + 2s)_{|l|−s}} P_{s}^{11}(k, l).

If we multiply (39) by 1/c^{|l|} and let c → ∞, we get

\frac{(−k_{1})_{l_{1}} ⋯ (−k_{n})_{l_{n}}}{(ν_{1})_{l_{1}} ⋯ (ν_{n})_{l_{n}}} = (−1)^{s} ∑_{s=0}^{|l|} \frac{(s − |k|)_{|l|−s}}{(|ν| + 2s)_{|l|−s}} P_{s}^{11}(k, l).

The case |k| = |l| of this equation reads

\frac{k_{1}! ⋯ k_{n}!}{(ν_{1})_{k_{1}} ⋯ (ν_{n})_{k_{n}}} δ_{k,l} = ∑_{s=0}^{|k|} \frac{(|k| − s)!}{(|ν| + 2s)_{|k|−s}} P_{s}^{11}(k, l), |k| = |l|.

This may also be deduced by inserting f = z_{1}^{k_{1}} ⋯ z_{n}^{k_{n}}, g = z_{1}^{l_{1}} ⋯ z_{n}^{l_{n}} in (38). For n = 2, writing P_{s}^{11}(k, l) as the product of two _{3}F_{2}-series, this gives the orthogonality relation for the dual Hahn polynomials (Eberlein polynomials).

8. Proof of Theorem 6.1

In this section we will prove the explicit expressions for coupling kernels given in Theorem 6.1. The proof will be based on equation (30). Thus we will first compute the matrix elements

(42)  ⟨Π_{s}z^{t}, z^{u}⟩_{ν}, |t| = |u| = s

of the projection Π_{s}.

We will work with the action of the universal enveloping algebra U(𝔰𝔩(2, ℂ)) derived from the group action. This associative algebra has three generators E, F, and H. The action is given on A^{ν} by the densely defined operators (cf. [R])

E = −\frac{d}{dz} , F = z^{2} \frac{d}{dz} + νz, H = − (2z \frac{d}{dz} + ν)

and on A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} by

E = − ∑_{i=1}^{n} \frac{∂}{∂z_{i}} , F = ∑_{i=1}^{n} (z_{i}^{2} \frac{∂}{∂z_{i}} + ν_{i}z_{i}) , H = − ∑_{i=1}^{n} (2z_{i} \frac{∂}{∂z_{i}} + ν_{i}) .

These operators satisfy the structure equations

[EF] = H, [HE] = 2E, [HF] = −2F

and the reality conditions

E^{∗} = −F, F^{∗} = −E, H^{∗} = H.

Let H_{s} be the subspace of A^{ν_{1}} ⊗⋯⊗A^{ν_{n}} consisting of homogeneous polynomials of degree s and let, as above, V_{s} be the subspace consisting of highest weight polynomials of degree s. There is then an orthogonal decomposition

H_{s} = V_{s} ⊕ F H_{s−1},

and we are interested in the orthogonal projection Π_{s}|_{H_{s}} onto the first summand.

Lemma 8.1. We have

EF^{k}|_{H_{s}} = F^{k}E − k(|ν| + 2s + k − 1)F^{k−1}.

Proof. First we observe that

EF |_{H_{s}} = (F E + H)|_{H_{s}} = F E − (|ν| + 2s)Id.

Proceeding by induction on k, we find that

EF^{k+1}|_{H_{s}} = EF^{k}|_{H_{s+1}}F |_{H_{s}} = F^{k}EF − k(|ν| + 2s + k + 1)F^{k}
= F^{k}(F E − (|ν| + 2s)) − k(|ν| + 2s + k + 1)F^{k}
= F^{k+1}E − (k + 1)(|ν| + 2s + k)F^{k}.
□

ORTHOGONAL POLYNOMIALS AND COUPLING COEFFICIENTS 27

Lemma 8.2. The orthogonal projection of H_{s} onto V_{s} is given by

∑_{k=0}^{∞} \frac{(−1)^{k}}{k!(2 − |ν| − 2s)_{k}} F^{k}E^{k}.

Note that since E^{s+1}|_{H_{s}} = 0, the sum terminates. Proof. We first show that the operator maps into V_{s}. Using the previous lemma, we find that in general

E ∑_{k=0}^{∞} c_{k}F^{k}E^{k}|_{H_{s}} = ∑_{k=0}^{∞} c_{k}EF^{k}|_{H_{s−k}}E^{k}
= ∑_{k=0}^{∞} c_{k}(F^{k}E − k(|ν| + 2s − k − 1)F^{k−1})E^{k}
= ∑_{k=0}^{∞}(c_{k} − c_{k+1}(k + 1)(|ν| + 2s − k − 2))F^{k}E^{k+1}.

This vanishes if

c_{k+1} = \frac{1}{(k + 1)(|ν| + 2s − k − 2)}c_{k},

which is indeed solved by

c_{k} = \frac{(−1)^{k}}{k!(2 − |ν| − 2s)_{k}} .

We then have, for any f in H_{s},

f = ∑_{k=0}^{∞} \frac{(−1)^{k}}{k!(2 − |ν| − 2s)_{k}} F^{k}E^{k}f − ∑_{k=1}^{∞} \frac{(−1)^{k}}{k!(2 − |ν| − 2s)_{k}} F^{k}E^{k}f,

where the first term is in V_{s}. Since the second term is in the image of F, it is orthogonal to the first term. This completes the proof. □

We use this lemma to compute the matrix elements (42). Lemma 8.3. For |t| = |u| = s,

(43)  ⟨Π_{s}z^{t}, z^{u}⟩_{ν} = \frac{(−1)^{s}s!}{(|ν| + s − 1)_{s}} F_{1:1}^{1:2} ( \begin{matrix}|ν| + s − 1 & : & −t, −u \\ −s & : & ν \end{matrix} | 1) .

Proof. Since F^{∗} = −E, the previous lemma gives

(44)  ⟨Π_{s}z^{t}, z^{u}⟩_{ν} = ∑_{k=0}^{∞} \frac{1}{k!(2 − |ν| − 2s)_{k}} ⟨E^{k}z^{t}, E^{k}z^{u}⟩_{ν}.

Since E is the sum of the commuting operators −\frac{∂}{∂z_{j}}, it follows from the multinomial theorem that

E^{k}(z^{t}) = (−1)^{s}k! ∑_{|j|=s−k} \frac{(−t)_{j}}{j!} z^{j}

and thus that

⟨E^{k}z^{t}, E^{k}z^{u}⟩_{ν} = (k!)^{2} ∑_{|j|=s−k} \frac{(−t)_{j}(−u)_{j}}{j! (ν)_{j}} .

Inserting this in (44), we obtain

⟨Π_{s}z^{t}, z^{u}⟩_{ν} = ∑_{k=0}^{∞} ∑_{|j|=s−k} \frac{k! (−t)_{j}(−u)_{j}}{(2 − |ν| − 2s)_{k}j! (ν)_{j}}

= ∑_{j_{1},...,j_{n}=0}^{∞} \frac{(s − |j|)! (−t)_{j}(−u)_{j}}{(2 − |ν| − 2s)_{s−|j|}j! (ν)_{j}}

= \frac{(−1)^{s}s!}{(|ν| + s − 1)_{s}} ∑_{j_{1},...,j_{n}=0}^{∞} \frac{(|ν| + s − 1)_{|j|}(−t)_{j}(−u)_{j}}{(−s)_{|j|}j! (ν)_{j}}

= \frac{(−1)^{s}s!}{(|ν| + s − 1)_{s}} F_{1:1}^{1:2} ( \begin{matrix}|ν| + s − 1 & : & −t, −u \\ −s & : & ν \end{matrix} | 1) .
□

We are now ready to prove Theorem 6.1.

Proof of Theorem 6.1. Insert the expression (43) in (30). In the case of P_{s}^{12} this gives

P_{s}^{12}(k, ξ) = ∑_{|t|=|u|=s} \frac{(ν)_{t}(ν)_{u}}{t! u!} ⟨Π_{s}z^{t}, z^{u}⟩_{ν}(−1)^{s} \frac{(−k)_{t} ξ^{u}}{(ν)_{t}(ν)_{u}}

= ∑_{|t|=|u|=s} \frac{s!}{(|ν| + s − 1)_{s}} F_{1:1}^{1:2} ( \begin{matrix}|ν| + s − 1 & : & −t, −u \\ −s & : & ν \end{matrix} | 1) \frac{(−k)_{t} ξ^{u}}{t! u!}

= \frac{s!}{(|ν| + s − 1)_{s}} ∑_{|t|=|u|=s} ∑_{j_{1},...,j_{n}=0}^{∞}\frac{(|ν| + s − 1)_{|j|}(−t)_{j}(−u)_{j}(−k)_{t} ξ^{u}}{(−s)_{|j|}(ν)_{j}j! t! u!} .

Since the terms for which t_{i} ≤ j_{i} or u_{i} ≤ j_{i} for some i vanish, we put

t = j + p, u = j + q

and change the order of summation. Then

\frac{(−t)_{j}(−u)_{j}}{t! u!} = \frac{(−j − p)_{j}(−j − q)_{j}}{(j + p)! (j + q)!} = \frac{1}{p! q!}

so that we get

P_{s}^{12}(k, ξ) (45)

= \frac{s!}{(|ν| + s − 1)_{s}} ∑_{j: |j|≤s} \frac{(|ν| + s − 1)_{|j|}}{(−s)_{|j|}(ν)_{j}j!} ∑_{|p|=s−|j|} \frac{(−k)_{j+p}}{p!} ∑_{|q|=s−|j|} \frac{ξ^{j+q}}{q!} .

Now, by the generalized Chu–Vandermonde formula,

∑_{|p|=s−|j|} \frac{(−k)_{j+p}}{p!} = (−k)_{j} ∑_{|p|=s−|j|} \frac{(j − k)_{p}}{p!} = (−k)_{j} \frac{(|j| − |k|)_{s−|j|}}{(s − |j|)!}

(46)

= (−1)^{|j|} \frac{(−|k|)_{s}}{s!} \frac{(−s)_{|j|}(−k)_{j}}{(−|k|)_{|j|}} .

Similarly, the multinomial theorem gives

∑_{|q|=s−|j|} \frac{ξ^{j+q}}{q!} = ξ^{j} \frac{|ξ|^{s−|j|}}{(s − |j|)!} = (−1)^{|j|} \frac{|ξ|^{s}}{s!} \frac{(−s)_{|j|}ξ^{j}}{|ξ|^{|j|}} .

Inserting these expressions in (45) we finally obtain

P_{s}^{12}(k, ξ) = \frac{(−|k|)_{s}|ξ|^{s}}{(|ν| + s − 1)_{s}s!} ∑_{j_{1},...,j_{n}=0}^{∞} \frac{(|ν| + s − 1)_{|j|}(−s)_{|j|}(−k)_{j}ξ^{j}}{(−|k|)_{|j|}(ν)_{j}|ξ|^{|j|}}

= \frac{(−|k|)_{s}|ξ|^{s}}{(|ν| + s − 1)_{s}s!} F_{1:1}^{2:1} ( \begin{matrix}|ν| + s − 1, −s & : & −k \\ −|k| & : & ν \end{matrix} | \frac{ξ}{|ξ|}) .

The remaining two cases are similar, except that for P_{s}^{11} one uses the generalized Chu–Vandermonde formula twice and for P_{s}^{22} the multinomial theorem twice.

9. The Fock space

So far our discussion has been based on the unit disc. However, one may study analogous problems for other Hermitean symmetric spaces. In dimension 1 one has, apart from the disc, the Riemann sphere and the plane. For the Riemann sphere, the spaces A^{ν} should be replaced by finite-dimensional spaces of polynomials. Since the group SU(2) has only one conjugacy class of one-parameter subgroups, there will only be one analogue of the transforms T_{i}. This transform is similar to T_{1} with the parameters ν_{i} replaced by negative integers.

A more interesting case is the plane ℂ. We will write down the analogues of our results in this section. Since most proofs are similar to the case of the disc, we will not give the details. First we introduce the Fock space F_{α} (α > 0), consisting of entire functions for which

∥f∥^{2} = \frac{α}{π} ∫_{ℂ} |f(z)|^{2}e^{−α|z|^{2}} dxdy = ∑_{k=0}^{∞} \frac{k!}{α^{k}} |f̂(k)|^{2} < ∞.

This space may be viewed as a limit of A^{ν} when ν → ∞. In fact, if ν > 1, the norm of A^{ν} is given by

∥f∥^{2} = \frac{ν − 1}{π} ∫_{D} |f(z)|^{2}(1 − |z|^{2})^{ν−2} dxdy.

If we replace D with the disc {|z| < R} and let ν and R tend to infinity so that ν/R^{2} → α, the weight tends to

lim_{ν, R→∞} (1 − \frac{|z|^{2}}{R^{2}} )^{ν−2} = e^{−α|z|^{2}}.

At the level of Lie algebras, this corresponds to deforming 𝔰𝔲(1, 1) into the Heisenberg algebra.

On the space F_{α} we introduce the Heisenberg operators

U_{h}f(z) = e^{−α|h|^{2}/2−αh̄z}f(z + h), h ∈ ℂ.

They satisfy the Heisenberg relation

U_{h_{1}}U_{h_{2}} = e^{iαIm(h_{1}h̄_{2})}U_{h_{1}+h_{2}}

and are unitary operators.

As an analogue of the parabolic basis for A^{ν} we introduce the Gauss–Weierstrass functions

e_{x}^{2}(z) = e^{−αz^{2}/2+xz},

and write 𝔉_{2} for the transform

𝔉_{2}f(x) = ⟨f, e_{x}^{2}⟩.

We also write e_{k}^{1}(z) = z^{k} and

𝔉_{1}f(k) = ⟨f, e_{k}^{1}⟩.

In analogy with (13), and with the same interpretation, we have the Plancherel formulas

(47)  ⟨f, g⟩ = ∑_{k=0}^{∞} \frac{α^{k}}{k!} 𝔉_{1}f(k) \overline{𝔉_{1}g(k)}

= \frac{1}{\sqrt{2πα}} ∫_{−∞}^{∞} 𝔉_{2}f(x) \overline{𝔉_{2}g(x)} e^{−x^{2}/2α} dx.

The matrix elements of U_{h} are given by

⟨U_{h}e_{k}^{1}, e_{l}^{1}⟩ = (−1)^{l}h^{k}h̄^{l}e^{−α|h|^{2}/2} _{2}F_{0} ( \begin{matrix}−k, −l \\ − \end{matrix} | − \frac{1}{α|h|^{2}} ) ,

which are essentially Charlier polynomials, and by

⟨e_{k}^{1}, e_{x}^{2}⟩ = \frac{x^{k}}{α^{k}} _{2}F_{0} ( \begin{matrix}−\frac{k}{2}, \frac{1}{2} − \frac{k}{2} \\ − \end{matrix} | − \frac{2α}{x^{2}})
= ∑_{0≤j≤[\frac{k}{2}]} \frac{(−1)^{j}k! x^{k−2j}}{(k − 2j)! j! 2^{j}α^{k−j}} ,

which are essentially Hermite polynomials. It will be convenient to denote the latter quantity by H_{k}(x, α), so that

(48)  e_{x}^{2}(z) = ∑_{k=0}^{∞} \frac{α^{k}}{k!} H_{k}(x, α)z^{k}.

In the standard notation for Hermite polynomials, cf. [VK], one has

H_{k}(x, α) = \frac{1}{(2α)^{k/2}} H_{k} (\frac{x}{\sqrt{2α}}) , α > 0.

The operators U_{h} permute the functions e_{x}^{2}, and thus the matrix elements ⟨U_{h}e_{k}^{1}, e_{x}^{2}⟩ are also given by Hermite polynomials. For the same reason, the matrix elements ⟨U_{h}e_{x}^{2}, e_{y}^{2}⟩ are essentially Dirac measures. We will also consider the matrix elements ⟨δ_{c}e_{x}^{2}, e_{y}^{2}⟩, where, as above, δ_{c} is the dilatation (δ_{c}f)(z) = f(cz). Expanding the scalar product as in (47), we get

(49)  ⟨δ_{c}e_{x}^{2}, e_{y}^{2}⟩ = ∑_{k=0}^{∞} \frac{α^{k}c^{k}}{k!} H_{k}(x, α)H_{k}(y, α).

Mehler’s formula, cf. [VK], equation 9.6.8(5), then gives

(50)  ⟨δ_{c}e_{x}^{2}, e_{y}^{2}⟩ = \frac{1}{\sqrt{1 − c^{2}}} exp (\frac{2cxy − c^{2}(x^{2} + y^{2})}{2α(1 − c^{2})}) , |c| < 1

(cf. [P1] for another computation of this scalar product).

Later we will need power series expansions of 1/e_{x}^{2} and 1/⟨δ_{c}e_{x}^{2}, e_{y}^{2}⟩. Since

\frac{1}{e_{x}^{2}(z)} = e^{αz^{2}/2−xz}

is obtained from e_{x}^{2}(z) by replacing α with −α and z with −z, it follows from (48) that

(51)  \frac{1}{e_{x}^{2}(z)} = ∑_{k=0}^{∞} \frac{α^{k}}{k!} H_{k}(x, −α)z^{k}.

Similarly, if in the right-hand side of (50) we formally replace α by −α and multiply by 1−c^{2}, we obtain the reciprocal quantity. Manipulating (49) in the same way gives

\frac{1}{⟨δ_{c}e_{x}^{2}, e_{y}^{2}⟩} = (1 − c^{2}) ∑_{k=0}^{∞} \frac{(−1)^{k}α^{k}c^{k}}{k!} H_{k}(x, −α)H_{k}(y, −α)

(52)

= ∑_{k=0}^{∞} \frac{(−1)^{k}α^{k}c^{k}}{k!} G_{k}(x, y, α),

where

G_{k}(x, y, α) = H_{k}(x, −α)H_{k}(y, −α) − \frac{k(k − 1)}{α^{2}}H_{k−2}(x, −α)H_{k−2}(y, −α)

(53)

= ∑_{0≤i,j≤[\frac{k}{2}]} \frac{k! (k − 2)! (k(k − 1) − 4ij)x^{k−2i}y^{k−2j}}{(k − 2i)! (k − 2j)! i! j! 2^{i+j}α^{2k−i−j}} .

Let us now consider an arbitrary tensor product

F_{α_{1}} ⊗ ⋯ ⊗ F_{α_{n}}

of Fock spaces. It decomposes under the group action into infinitely many copies of the space F_{|α|}. To get a finer decomposition one may add to the group auxiliary operators of the form

f(z) ↦ e^{imθ}f(e^{iθ}z),

where m is an integer; cf. [VK], Section 8.6.1. Temporarily writing F_{α,m} for the corresponding representation space, we have a decomposition

F_{α_{1},m_{1}} ⊗ ⋯ ⊗ F_{α_{n},m_{n}} = ⨁_{s=0}^{∞} \binom{n + s − 2}{n − 2}F_{|α|,|m|+s}.

This means that the intertwining embeddings F_{|α|,|m|+s} → F_{α_{1},m_{1}} ⊗ ⋯ ⊗ F_{α_{n},m_{n}} are required to increase homogeneity by s. If Q is an image of 1 under such an embedding, then Q must satisfy the same condition (3) as in the case of the disc. In the case at hand, there is a closed expression for the corresponding embedding K_{Q}:

(54)  (K_{Q}g)(z_{1}, …, z_{n}) = Q(z_{1}, …, z_{n}) g (\frac{(α, z)}{|α|}) ,

where

(α, z) = α_{1}z_{1} + ⋯ + α_{n}z_{n}.

¿From now on we write

⟨f, g⟩_{α} = ⟨f, g⟩_{F_{α_{1}}⊗···⊗F_{αn}}, ⟨f, g⟩_{|α|} = ⟨f, g⟩_{F_{|α|}}.

To proceed in analogy with the case of the disc, we should now seek factorizations of the transforms

𝔉_{i}^{h} = 𝔉_{i} U_{h}, i = 1, 2.

However, we get a useful factorization only in the case i = 1. Then we may write

𝔉_{1}^{h} = M_{1} 𝔗_{1} δ_{−1/h̄} τ_{h} (h ≠ 0),

where

M_{1}f(k) = e^{−α|h|^{2}/2}(−1)^{k}h̄^{k}f(k)

and

𝔗_{1}z^{k}(m) = (−1)^{k} \frac{(−m)_{k}}{α^{k}} .

In analogy with Theorem 4.2, we have

(55)  (𝔉_{1}^{h} K_{Q}g)(m) = 𝔗_{1}Q(m)(𝔉_{1}^{h}g)(|m| − s).

Thus if

Q(z) = ∑_{|t|=s} c_{t} z_{1}^{t_{1}} ⋯ z_{n}^{t_{n}}

is a highest weight vector, we have a corresponding coupling coefficient

𝔗_{1}Q(m_{1}, …, m_{n}) = (−1)^{s} ∑_{|t|=s} c_{t} \frac{(−m_{1})_{t_{1}} ⋯ (−m_{n})_{t_{n}}}{α_{1}^{t_{1}} ⋯ α_{n}^{t_{n}}} ,

which is again a polynomial of degree s.

For i = 2 we define

𝔗_{2}Q(x_{1}, …, x_{n}) = ⟨Q, e_{x_{1}}^{2} ⊗ ⋯ ⊗ e_{x_{n}}^{2}⟩_{α},

where Q is a highest weight vector. There is no very simple expression for 𝔗_{2}Q; we must be content with writing

𝔗_{2}Q(x_{1}, …, x_{n}) = ∑_{|t|=s} c_{t} H_{t_{1}}(x_{1}, α_{1}) ⋯ H_{t_{n}}(x_{n}, α_{n}),

which shows that 𝔗_{2}Q is a polynomial of degree s. One may also show that

(56)  𝔗_{2}Q(x_{1} + bα_{1}, …, x_{n} + bα_{n}) = 𝔗_{2}Q(x_{1}, …, x_{n}), b ∈ ℂ.

Note that in the case of the disc, the polynomial T_{2}Q is homogeneous. Thus in both cases something of the property (3) is preserved. As in Theorem 4.2, we have

(57)  (𝔉_{2}^{h} K_{Q}g)(x) = 𝔗_{2}Q(x)(𝔉_{2}^{h}g)(|x|)

and, for the dilatations,

(58)  (𝔉_{2}δ_{c}K_{Q}g)(x) = c^{s}𝔗_{2}Q(x)(𝔉_{2}δ_{c}g)(|x|).

In the bilinear case, n = 2, we may take Q(z) = (z_{1}−z_{2})^{s}. Then 𝔗_{1}Q are essentially Krawtchouk polynomials; in the notation of [VK] we have

𝔗_{1}Q(m_{1}, m_{2}) = \frac{(−m_{1} − m_{2})_{s}}{α_{2}^{s}} K_{s}(m_{1}; \frac{α_{1}}{α_{1}+α_{2}} ; m_{1} + m_{2}),

while the polynomials 𝔗_{2}Q may be expressed in terms of Hermite polynomials. Indeed, we may write

𝔗_{2}Q(x_{1}, x_{2}) = ∑_{k=0}^{s} \binom{s}{k}(−1)^{s−k}H_{k}(x_{1}, α_{1})H_{s−k}(x_{2}, α_{2})

= (\frac{1}{α_{1}} + \frac{1}{α_{2}} )^{s} H_{s} (\frac{x_{1}}{α_{1}} − \frac{x_{2}}{α_{2}} , \frac{1}{α_{1}} + \frac{1}{α_{2}} ) ,

where the last equality is a special case of the addition formula for Hermite polynomials, cf. [VK], equation 9.6.8(7), which is in turn easily derived from (48). Note that this expression is in agreement with (56). That Krawtchouk polynomials are Clebsch–Gordan coefficients for the Heisenberg algebra is well-known, cf. [VK], while for the Hermite polynomial this fact, as far as we know, first occurs in [VdJ].

We now write down some properties of the coupling coefficients. First we have the Clebsch–Gordan type formulas analogous to Corollary 4.3. In this case we may use (54) to write them more explicitly as

Q(z)\frac{(α, z)^{k}}{k!} = ∑_{|m|=k+s} \frac{α_{1}^{m_{1}} ⋯ α_{n}^{m_{n}}}{m_{1}! ⋯ m_{n}!} 𝔗_{1}Q(m) z_{1}^{m_{1}} ⋯ z_{n}^{m_{n}},

Q(z) exp (−\frac{(x − (α, z))^{2}}{2|α|} ) = \frac{1}{(2π)^{\frac{n−1}{2}}} \sqrt{\frac{|α|}{α_{1} ⋯ α_{n}}}

× ∫_{y∈ℝ^{n}: |y|=x} 𝔗_{2}Q(y) exp (− (\frac{(y_{1} − α_{1}z_{1})^{2}}{2α_{1}} + ⋯ + \frac{(y_{n} − α_{n}z_{n})^{2}}{2α_{n}} )) dy,

where in the last equality we express the product of the generalized basis elements and the weight of the Plancherel measures as

e_{y}^{2}(z) e^{−y^{2}/2α} = e^{−(y−αz)^{2}/2α}.

Next we have orthogonality relations as in Corollary 4.4. If Q and Q̃ are two highest weight vectors, then their scalar product ⟨Q, Q̃⟩_{α} equals both the sum

\frac{k!}{|α|^{k}} ∑_{|m|=k+s} \frac{α_{1}^{m_{1}} ⋯ α_{n}^{m_{n}}}{m_{1}! ⋯ m_{n}!} 𝔗_{1}Q(m) \overline{𝔗_{1}Q̃(m)}

for each k = 0, 1, 2, … and, for each x ∈ ℝ, the integral

\frac{1}{(2π)^{\frac{n−1}{2}}} \sqrt{\frac{|α|}{α_{1} ⋯ α_{n}}} ∫_{y∈ℝ^{n}: |y|=x} 𝔗_{2}Q(y) \overline{𝔗_{2}Q̃(y)} exp (− (\frac{y_{1}^{2}}{2α_{1}} + ⋯ + \frac{y_{n}^{2}}{2α_{n}} )) dy.

There are also analogues of the convolution formulas of Theorem 5.1. If Q is a highest weight polynomial of degree s, we have

\frac{|α|^{k}}{k!} c^{s}_{2}F_{0} ( \begin{matrix}−k, s − |l| \\ − \end{matrix} | \frac{c}{|α|}) 𝔗_{1}Q(l)

= ∑_{|m|=k+s} \frac{α_{1}^{m_{1}} ⋯ α_{n}^{m_{n}}}{m_{1}! ⋯ m_{n}!} 𝔗_{1}Q(m) ∏_{r=1}^{n}_{2}F_{0} ( \begin{matrix}−m_{r}, −l_{r} \\ − \end{matrix} | \frac{c}{α_{r}} ) ,

\frac{|α|^{k}}{k!} H_{k}(|x|, |α|) 𝔗_{2}Q(x)

= ∑_{|m|=k+s} \frac{α_{1}^{m_{1}} ⋯ α_{n}^{m_{n}}}{m_{1}! ⋯ m_{n}!} 𝔗_{1}Q(m) ∏_{r=1}^{n} H_{m_{r}}(x_{r}, α_{r}),

exp (−\frac{x^{2}}{2|α|}) H_{|m|−s}(x, |α|) 𝔗_{1}Q(m) = \frac{1}{(2π)^{\frac{n−1}{2}}} \sqrt{\frac{|α|}{α_{1} ⋯ α_{n}}}

× ∫_{y∈ℝ^{n}: |y|=x} 𝔗_{2}Q(y) ∏_{r=1}^{n} H_{m_{r}}(y_{r}, α_{r}) exp (− (\frac{y_{1}^{2}}{2α_{1}} + ⋯ + \frac{y_{n}^{2}}{2α_{n}} )) dy,

c^{s} exp (\frac{2cx|z| − x^{2} − c^{2}|z|^{2}}{2|α|(1 − c^{2})} ) 𝔗_{2}Q(z) = \frac{1}{(2π(1 − c^{2}))^{\frac{n−1}{2}}} \sqrt{\frac{|α|}{α_{1} ⋯ α_{n}}}

× ∫_{y∈ℝ^{n}: |y|=x} 𝔗_{2}Q(y) exp (∑_{j=1}^{n} \frac{2cy_{j}z_{j} − y_{j}^{2} − c^{2}z_{j}^{2}}{2α_{j}(1 − c^{2})} ) dy.

For n = 2, the first of these formulas is the addition formula for Charlier polynomials, cf. [VK], Section 8.6.5, while the second one is the generalization of the addition formula for Hermite polynomials given in [VdJ] and [KV1]. As degenerate cases of the first of these formulas we obtain analogues of the equations (26) and (27):

\frac{|α|^{k+s}}{(k + s)!} 𝔗_{1}Q(l) = ∑_{|m|=k+s} \frac{α_{1}^{m_{1}} ⋯ α_{n}^{m_{n}}}{m_{1}! ⋯ m_{n}!} 𝔗_{1}Q(m + l),

\frac{(s − |l|)_{k}}{k!} 𝔗_{1}Q(l) = (−1)^{s} ∑_{|m|=k+s} \frac{(−l_{1})_{m_{1}} ⋯ (−l_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} 𝔗_{1}Q(m).

They are proved similarly using the limit

lim_{c→∞} \frac{1}{c^{m}} _{2}F_{0} ( \begin{matrix}−m, −l \\ − \end{matrix} | \frac{c}{a}) = (−1)^{m} \frac{(−l)_{m}}{a^{m}} (m, l = 0, 1, 2, …).

We now turn to the coupling kernels for the present situation. They are defined as in the case of the disc and have a similar group theoretic interpretation. We denote them by 𝔓_{s}^{ij}. We may reduce ourselves to the three cases

𝔓_{s}^{11}(k, l),  𝔓_{s}^{12}(k, x),  𝔓_{s}^{22}(x, y).

We first give explicit expressions for the coupling kernels similar to those given in Theorem 6.1. It seems that the kernels 𝔓_{s}^{12} and 𝔓_{s}^{22} do not have simple expressions in terms of Karlsson’s functions. Instead we give expansions involving Hermite polynomials. Our explicit expressions are

(59) 𝔓_{s}^{11}(k, l) = \frac{(−1)^{s}(−|k|)_{s}(−|l|)_{s}}{|α|^{s}s!} F_{2:0}^{1:2} ( \begin{matrix}−s & : & −k, −l \\ −|k|, −|l| & : & − \end{matrix} | \frac{|α|}{α}) ,
𝔓_{s}^{12}(k, x) =

= \frac{(−|k|)_{s}}{|α|^{s}} ∑_{m: |m|≤s} \frac{(−1)^{|m|}|α|^{|m|}(−k)_{m}}{(−|k|)_{|m|}m!} ∑_{|p|=s−|m|} \frac{α^{p}}{p!} H_{p+m}(x, α) (60)

= \frac{(−1)^{s}(−|k|)_{s}}{s!} ∑_{m_{1},...,m_{n}=0}^{∞}(−1)^{|m|} \frac{(−s)_{|m|}(−k)_{m}}{(−|k|)_{|m|}m!} H_{s−|m|}(|x|, −|α|)H_{m}(x, α), (61)

𝔓_{s}^{22}(x, y) =

= \frac{(−1)^{s}s!}{|α|^{s}} ∑_{m: |m|≤s} \frac{|α|^{|m|}α^{m}}{(−s)_{|m|}m!} ∑_{|p|=s−|m|} \frac{α^{p}}{p!} H_{p+m}(x, α) ∑_{|q|=s−|m|} \frac{α^{q}}{q!} H_{q+m}(y, α) (62)

= \frac{(−1)^{s}|α|^{s}}{s!} ∑_{m_{1},...,m_{n}=0}^{∞} \frac{(−s)_{|m|}α^{m}}{|α|^{|m|}m!} G_{s−|m|}(|x|, |y|, |α|)H_{m}(x, α)H_{m}(y, α), (63)

where in (59) we write for short

\frac{|α|}{α} = (\frac{α_{1} + ⋯ + α_{n}}{α_{1}} , …, \frac{α_{1} + ⋯ + α_{n}}{α_{n}} ) ,

where

H_{m}(x, α) = H_{m_{1}}(x_{1}, α_{1}) ⋯ H_{m_{n}}(x_{n}, α_{n}),

and where G is the polynomial defined in (53).

As for the proof, the expression (59) may be proved in a similar manner as in the case of the disc, using instead of Lemma 8.2 the formula

Π_{s}|_{H_{s}} = ∑_{k=0}^{∞} \frac{1}{k!|α|^{k}} F^{k}E^{k},

where Π_{s} is defined as the orthogonal projection of F_{α_{1}} ⊗⋯⊗F_{α_{n}} onto the subspace spanned by all K_{Q}g, where g ∈ F_{|α|} and Q is a highest weight polynomial of degree s and where

E = − ∑_{i=1}^{n} \frac{∂}{∂z_{i}} , F = ∑_{i=1}^{n} α_{i}z_{i}

are Lie algebra operators derived from the group action. If one continues to imitate the proof of Theorem 6.1, one arrives at the expressions (60) and (62). However, in this case there is no “multinomial theorem” to sum the inner series. To prove (61) and (63), which we believe are the natural analogues of the expressions in Theorem 6.1, we use a different approach.

Proof of (61). Consider a tensor product e_{x_{1}}^{2} ⊗ ⋯ ⊗ e_{x_{n}}^{2}. We try to decompose it as

e_{x_{1}}^{2} ⊗ ⋯ ⊗ e_{x_{n}}^{2} = ∑_{s=0}^{∞} K_{Q_{sx}}g_{sx},

where Q_{sx} is a highest weight polynomial of degree s. Because of (57) we expect that one must take g_{sx} = e_{|x|}^{2} for all s, and thus, in view of (54), that

e_{x_{1}}^{2}(z_{1}) ⋯ e_{x_{n}}^{2}(z_{n}) = e_{|x|}^{2} (\frac{(α, z)}{|α|}) ∑_{s=0}^{∞} Q_{sx}.

That there is such an expansion may be verified easily by checking that the function

(64)  \frac{e_{x_{1}}^{2}(z_{1}) ⋯ e_{x_{n}}^{2}(z_{n})}{e_{|x|}^{2} (\frac{(α,z)}{|α|})} = exp (∑_{i=1}^{n} (−\frac{α_{i}}{2} z_{i}^{2} + x_{i}z_{i}) + \frac{(α, z)^{2}}{2|α|} − \frac{|x|(α, z)}{|α|})

is in the kernel of E and thus that its homogeneous parts are highest weight polynomials. This may be made more transparent by rewriting the above expression as

\frac{e_{x_{1}}^{2}(z_{1}) ⋯ e_{x_{n}}^{2}(z_{n})}{e_{|x|}^{2} (\frac{(α,z)}{|α|})} = exp (−\frac{1}{2|α|} ∑_{i<j} α_{i}α_{j}(z_{i} − z_{j})^{2} + 2(x_{i}α_{j} − α_{i}x_{j})(z_{i} − z_{j})) .

Now, consider the matrix element

⟨Π_{s}e_{x}^{2}, e_{k}^{1}⟩_{α} = ⟨Π_{s} e_{x_{1}}^{2} ⊗ ⋯ ⊗ e_{x_{n}}^{2}, e_{k_{1}}^{1} ⊗ ⋯ ⊗ e_{k_{n}}^{1}⟩_{α}

for Π_{s}. As for the disc, it factors as

⟨Π_{s}e_{x}^{2}, e_{k}^{1}⟩_{α} = 𝔓_{s}^{12}(k, x)⟨e_{|x|}^{2}, e_{|k|−s}^{1}⟩_{|α|}.

On the other hand, (55) gives

⟨Π_{s}e_{x}^{2}, e_{k}^{1}⟩_{α} = ⟨K_{Q_{x}^{s}}e_{|x|}^{2}, e_{k}^{1}⟩_{α} = 𝔗_{1}Q_{sx}(k)⟨e_{|x|}^{2}, e_{|k|−s}^{1}⟩_{|α|}.

Thus we have found that

𝔓_{s}^{12}(k, x) = 𝔗_{1}Q_{sx}(k),

where Q_{sx} is the term of degree s in the homogeneous expansion of the function (64). Now, by (48)

e_{x_{1}}^{2}(z_{1}) ⋯ e_{x_{n}}^{2}(z_{n}) = ∑_{m_{1},...,m_{n}=0}^{∞} \frac{α^{m}z^{m}}{m!} H_{m}(x, α),

and by (51)

\frac{1}{e_{|x|}^{2} (\frac{(α,z)}{|α|})} = ∑_{l=0}^{∞} \frac{(α, z)^{l}}{l!} H_{l}(|x|, −|α|).

Thus

Q_{sx}(z) = ∑_{m: |m|≤s} \frac{α^{m}z^{m}}{m!} H_{m}(x, α) \frac{(α, z)^{s−|m|}}{(s − |m|)!} H_{s−|m|}(|x|, −|α|).

Expanding the power (α, z)^{s−|m|} gives

Q_{sx}(z) = ∑_{|m|+|p|=s} \frac{α^{m+p}z^{m+p}}{m! p!} H_{m}(x, α)H_{s−|m|}(|x|, −|α|).

By the definition of 𝔗_{1}, it follows that

𝔗_{1}Q_{sx}(k) = (−1)^{s} ∑_{|m|+|p|=s} \frac{(−k)_{m+p}}{m! p!} H_{m}(x, α)H_{s−|m|}(|x|, −|α|).

Summing in p, we have as in (46) that

∑_{|p|=s−|m|} \frac{(−k)_{m+p}}{p!} = (−1)^{|m|} \frac{(−|k|)_{s}}{s!} \frac{(−s)_{|m|}(−k)_{m}}{(−|k|)_{|m|}} .

Inserting this in the previous formula gives the desired expression (61).

Proof of (63). We consider the matrix element

⟨δ_{c}e_{x}^{2}, e_{y}^{2}⟩_{α} = ⟨(δ_{c})^{⊗n}e_{x_{1}}^{2} ⊗ ⋯ ⊗ e_{x_{n}}^{2}, e_{y_{1}}^{2} ⊗ ⋯ ⊗ e_{y_{n}}^{2}⟩_{α}

of δ_{c}, |c| < 1, and expand it as

⟨δ_{c}e_{x}^{2}, e_{y}^{2}⟩_{α} = ∑_{s=0}^{∞}⟨Π_{s}δ_{c}e_{x}^{2}, e_{y}^{2}⟩_{α}.

By (58)

⟨Π_{s}δ_{c}e_{x}^{2}, e_{y}^{2}⟩_{α} = c^{s} 𝔓_{s}^{22}(x, y)⟨δ_{c}e_{|x|}^{2}, e_{|y|}^{2}⟩_{|α|}.

It follows that 𝔓_{s}^{22}(x, y) is the coefficient of c^{s} in

\frac{⟨δ_{c}e_{x}^{2}, e_{y}^{2}⟩_{α}}{⟨δ_{c}e_{|x|}^{2}, e_{|y|}^{2}⟩_{|α|}} .

Using the expansions (48) and (52) one obtains the expression (63).

For n = 2, the coupling kernels factor as products of two Clebsch–Gordan coefficients. Choosing Q(z) = (z_{1} − z_{2})^{s}, we may write

𝔓_{s}^{ij}((x_{1}, x_{2}), (y_{1}, y_{2})) = \frac{1}{∥Q∥_{α}^{2}} 𝔗_{i}Q(x_{1}, x_{2}) 𝔗_{j}Q(y_{1}, y_{2}), i, j = 1, 2,

where the norm is easily computed:

∥Q∥_{α}^{2} = s! (\frac{1}{α_{1}} + \frac{1}{α_{2}} )^{s} .

Comparing with the expressions for 𝔓_{s}^{ij} given above leads to Watson-type formulas, which we have not found in the literature. The case i = j = 1 gives the identity

\frac{α_{1}^{s}}{α_{2}^{s}} K_{s}(k_{1}; \frac{α_{1}}{α_{1}+α_{2}} ; k_{1} + k_{2}) K_{s}(l_{1}; \frac{α_{1}}{α_{1}+α_{2}} ; l_{1} + l_{2})

= (−1)^{s} F_{2:0}^{1:2} ( \begin{matrix}−s & : & −(k_{1}, k_{2}), −(l_{1}, l_{2}) \\ −k_{1} − k_{2}, −l_{1} − l_{2} & : & − \end{matrix} | \frac{α_{1} + α_{2}}{α_{1}} , \frac{α_{1} + α_{2}}{α_{2}} ) .

Changing the names of the variables, this may be rewritten as

2F_{1} ( \begin{matrix}−n, a \\ b \end{matrix} | x)_{2}F_{1} ( \begin{matrix}−n, c \\ d \end{matrix} | x)

= (1 − x)^{n} F_{2:0}^{1:2} ( \begin{matrix}−n & : & (a, b − a), (c, d − c) \\ b, d & : & − \end{matrix} | x, \frac{x}{x − 1}) ,

which is true for arbitrary a, b, c, d since the series terminate. The case i = 1, j = 2 gives

\frac{1}{α_{2}^{s}} K_{s}(k_{1}; \frac{α_{1}}{α_{1}+α_{2}} ; k_{1} + k_{2})H_{s} (\frac{x_{1}}{α_{1}} − \frac{x_{2}}{α_{2}} , \frac{1}{α_{1}} + \frac{1}{α_{2}} )

= (−1)^{s} ∑_{m_{1},m_{2}=0}^{∞}(−1)^{m_{1}+m_{2}} \frac{(−s)_{m_{1}+m_{2}}(−k_{1})_{m_{1}}(−k_{2})_{m_{2}}}{(−k_{1} − k_{2})_{m_{1}+m_{2}}m_{1}! m_{2}!}
×H_{s−m_{1}−m_{2}}(x_{1} + x_{2}, −α_{1} − α_{2})H_{m_{1}}(x_{1}, α_{1})H_{m_{2}}(x_{2}, α_{2}).

Finally, the case i = j = 2 gives

H_{s} (\frac{x_{1}}{α_{1}} − \frac{x_{2}}{α_{2}} , \frac{1}{α_{1}} + \frac{1}{α_{2}} ) H_{s} (\frac{y_{1}}{α_{1}} − \frac{y_{2}}{α_{2}} , \frac{1}{α_{1}} + \frac{1}{α_{2}} )

= (−α_{1}α_{2})^{s} ∑_{m_{1},m_{2}=0}^{∞} \frac{(−s)_{m_{1}+m_{2}}α_{1}^{m_{1}}α_{2}^{m_{2}}}{(α_{1} + α_{2})^{m_{1}+m_{2}}m_{1}! m_{2}!} G_{s−m_{1}−m_{2}}(x_{1} + x_{2}, y_{1} + y_{2}, α_{1} + α_{2})
×H_{m_{1}}(x_{1}, α_{1})H_{m_{2}}(x_{2}, α_{2})H_{m_{1}}(y_{1}, α_{1})H_{m_{2}}(y_{2}, α_{2}).

We now discuss the biorthogonal polynomial systems analogous to the Appell and dual Appell polynomials of Sections 2.2 and 2.3. Thus we introduce the basis

Q_{t}(z) = (z_{1} − z_{n})^{t_{1}} ⋯ (z_{n−1} − z_{n})^{t_{n−1}}, t_{1} + ⋯ + t_{n−1} = s

of the space of highest weight vectors of degree s in F_{α_{1}} ⊗ ⋯ ⊗ F_{α_{n}} and let (Q′_{t}) denote the dual basis. As in the case of the disc, it is easy to find expressions for the polynomials 𝔗_{i}Q_{t} using the definition of 𝔗_{i} directly, while the polynomials 𝔗_{i}Q′_{t} arise as special cases of the coupling kernels; in fact

𝔗_{i}Q′_{t}(x) = \frac{α_{1}^{t_{1}} ⋯ α_{n−1}^{t_{n−1}}}{t_{1}! ⋯ t_{n−1}!} 𝔓_{s}^{1i}((t_{1}, …, t_{n−1}, 0), (x_{1}, …, x_{n})).

Writing as above x̂ = (x_{1}, …, x_{n−1}), where x = (x_{1}, …, x_{n}), we have

𝔗_{1}Q_{t}(m) = \frac{(−m_{n})_{s}}{α_{n}^{s}} F_{1:0}^{0:2} ( \begin{matrix}− & : & −t, −m̂ \\ 1 + m_{n} − s & : & − \end{matrix} | − \frac{α_{n}}{α_{1}} , …, −\frac{α_{n}}{α_{n−1}} ) ,
𝔗_{1}Q′_{t}(m) = \frac{α̂^{t}(−|m|)_{s}}{t! |α|^{s}} F_{1:0}^{0:2} ( \begin{matrix}− & : & −t, −m̂ \\ −|m| & : & − \end{matrix} | \frac{|α|}{α_{1}}, …, \frac{|α|}{α_{n−1}} ) ,

𝔗_{2}Q_{t}(x) = (−1)^{s} ∑_{k_{1},...,k_{n−1}=0}^{∞} \frac{(−t)_{k}}{k!} ∏_{j=1}^{n−1} H_{k_{j}}(x_{j}, α_{j})H_{s−|k|}(x_{n}, α_{n}),

𝔗_{2}Q′_{t}(x) = \frac{α̂^{t}}{t!} ∑_{k_{1},...,k_{n−1}=0}^{∞}(−1)^{|k|} \frac{(−t)_{k}}{k!} ∏_{j=1}^{n−1} H_{k_{j}}(x_{j}, α_{j})H_{s−|k|}(|x|, −|α|)

(Karlsson’s F_{1:0}^{0:2}-function equals Lauricella’s F_{B}-function). The first of these biorthogonal systems is Tratnik’s multivariable Krawtchouk polynomials [T3], while we have not found the second one in the literature.

Continuing this exposition of the Fock space analogues of our results, we should now proceed with Section 7. Special choices of Q in the orthogonality and convolution formulas above lead to analogues of Propositions 7.1, 7.2, and 7.3. The details of this are left to the reader. There are also analogues of the linearization formulas of Proposition 7.4. For 𝔓_{s}^{11} we have the linearization formula

∏_{r=1}^{n}_{2}F_{0} ( \begin{matrix}−k_{r}, −l_{r} \\ − \end{matrix} | \frac{c}{α_{r}} ) = ∑_{s=0}^{min(|k|,|l|)} 𝔓_{s}^{11}(k, l) c^{s}_{2}F_{0} ( \begin{matrix}s − |k|, s − |l| \\ − \end{matrix} | \frac{c}{|α|}) .

Degenerate cases of this formula are

\frac{(−k_{1})_{l_{1}} ⋯ (−k_{n})_{l_{n}}}{α_{1}^{l_{1}} ⋯ α_{n}^{l_{n}}} = (−1)^{s} ∑_{s=0}^{|l|} \frac{(s − |k|)_{|l|−s}}{|α|^{|l|−s}} 𝔓_{s}^{11}(k, l),

\frac{k_{1}! ⋯ k_{n}!}{α_{1}^{k_{1}} ⋯ α_{n}^{k_{n}}} δ_{k,l} = ∑_{s=0}^{|k|} \frac{(|k| − s)!}{|α|^{|k|−s}} 𝔓_{s}^{11}(k, l), |k| = |l|.

For 𝔓_{s}^{12} and 𝔓_{s}^{22} the linearization formulas seem to be of a more trivial nature. For 𝔓_{s}^{12} we obtain

∏_{r=1}^{n} H_{k_{r}}(x_{r}, α_{r}) = ∑_{s=0}^{|k|} 𝔓_{s}^{12}(k, x)H_{|k|−s}(|x|, |α|).

This is, however, very easy to deduce from (61). For 𝔓_{s}^{22}, the linearization formula is

∏_{r=1}^{n}⟨δ_{c}e_{x_{r}}, e_{y_{r}}⟩_{α_{r}} = ∑_{s=0}^{∞} c^{s} 𝔓_{s}^{22}(x, y)⟨δ_{c}e_{|x|}, e_{|y|}⟩_{|α|},

which we have already used in the proof of (63). As for the disc, when n = 2 we may factor 𝔓_{s}^{ij} as a product of two Clebsch–Gordan coefficients and obtain formulas of Burchnall–Chaundy-type. In the case of 𝔓_{s}^{11} we obtain the multiplication formula for Charlier polynomials in this way; cf. [VK], Section 8.6.5.

Acknowledgments

I am grateful to Jaak Peetre for his careful reading of this manuscript, leading to many improvements. I would also like to thank the referees for several useful comments.

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