ON THE TRIPLE SUM FORMULA FOR WIGNER 9j-SYMBOLS

HJALMAR ROSENGREN

Abstract. We give a new proof of the triple sum formula for Wigner 9j-symbols due to Ališauskas and Jucys. The proof uses explicit expressions for the coupling kernels recently introduced by the author. Parts of our results generalize to general recoupling coefficients.

1. Introduction

The Wigner 9j-symbols arise in quantum mechanics as recoupling coefficients of four angular momenta. They were first introduced by Eugene Wigner in 1940. For the general theory of Wigner symbols and related objects we refer to the two-volume treatise [BL1, BL2].

There are many known expressions for 9j-symbols as multiple hypergeometric sums. Generally speaking, the larger the number of summation variables, the more symmetries do these sums exhibit. The most compact known expression is the triple sum formula of Ališauskas and Jucys [AJ], which does not exhibit any of the 72 symmetries of the 9j-symbol. Because of the low number of summation variables, this formula is useful in numerical computations [SRC], and it has also been used to study the zeroes of 9j-symbols [SPV, SR1]. On the other hand, since the 72 symmetries are not exhibited in the triple sum formula, they give rise to nontrivial symmetries of this type of sums. This has been used to find several new hypergeometric summation and transformation formulas, cf. [PV, VPS1, VPS2].

The original proof in [AJ] uses a more complicated interpretation of 9j-symbols than the standard one, involving an embedding SU(2) × SU(2) ⊆ SO(5). An elementary proof was given in the book [JB]. There the starting point is a formula expressing the 9j-symbol as a double sum, each term being the product of three 3j-symbols (Clebsch–Gordan coefficients). The triple sum formula is then derived using only elementary series manipulations. The proof is however quite tricky, and it seems very hard to find without knowledge of the end result.

In a previous paper [R2], we introduced some polynomials in 2n variables which were called coupling kernels. They are the reproducing kernels of certain spaces of coupling coefficients for n angular momenta, and may be expressed as n-variable hypergeometric sums. Using this explicit formula (for n = 4) we will derive an expression for Wigner 9j-symbols as a double sum, each term being the product of only two 3j-symbols. After inserting appropriate single sum expressions for the 3j-symbols it is possible to perform the summation in one variable and arrive at a triple sum, which is equivalent to the one given in [AJ]. Thus there are no series manipulations involved and one can easily derive the formula without knowing it in advance.

1991 Mathematics Subject Classification. 22E70, 33C70, 81R05.

Key words and phrases. Wigner 9j-symbol, multiple hypergeometric function, highest weight representation.

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: J. Math. Phys. 39 (1998), 6730-6744.

We will follow the approach to coupling and recoupling coefficients of [R1, R2]. In particular, we will work with the Lie algebra 𝔰𝔲(1, 1) rather than with 𝔰𝔲(2), which is more common in the literature. It is well-known that the Wigner symbols of 𝔰𝔲(2) may be obtained from those of 𝔰𝔲(1, 1) by continuation. One may also generalize the relevant results of [R1, R2] to 𝔰𝔲(2), and then give a direct proof, analogous to the one below, of the triple sum formula in this case.

In Section 2 we survey parts of our previous papers [R1, R2]. Most of this is reformulation of well-known facts in slightly different terms. The only new result that we need is the explicit expression for coupling kernels from [R2] (equation (16) below). In Section 3 we prove an expansion formula, given in Theorem 3.1, of which our double sum mentioned above is a special case. In Section 4 we derive the Ališauskas–Jucys triple sum formula. Finally, in Section 5 we generalize Theorem 3.1 to arbitrary recoupling coefficients. We hope that this result, Theorem 5.2, may be useful also in the study of Wigner 3nj-symbols with n > 3.

2. Preliminaries

We will work with the holomorphic realization of highest weight representations of 𝔰𝔲(1, 1). Thus we let A^{ν} (ν > 0) be the Hilbert space of analytic functions in the complex unit disc with reproducing kernel (1 − zw̄)^{−ν}. The norm of a function

f(z) = ∑_{k=0}^{∞} a_{k}z^{k}

is then given by

∥f∥^{2} = ∑_{k=0}^{∞} \frac{k!}{(ν)_{k}} |a_{k}|^{2},

where

(ν)_{k} = ν(ν + 1) ⋯ (ν + k − 1), (ν)_{0} = 1

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

f ↦ f(ϕ(z))ϕ′(z)^{\frac{ν}{2}} ,

where ϕ is an analytic automorphism of the disc. The derived representation is a unitary representation of 𝔰𝔲(1, 1) on A^{ν} with the highest weight vector 1 of weight −ν. Passing to the complexified Lie algebra 𝔰𝔩(2, ℂ) yields the densely defined operators

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

which satisfy the structure equations

[EF] = H, [HE] = 2E, [HF] = −2F,

and the 𝔰𝔲(1, 1)-reality conditions

E^{∗} = −F, F^{∗} = −E, H^{∗} = H.

WIGNER 9j-SYMBOLS 3

Consider a Hilbert tensor product A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} of such spaces. We will use ∥ ∥ and ⟨ , ⟩ to denote its norm and scalar product. It decomposes under the Lie algebra action as

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

(where |ν| = ∑ ν_{i}), that is, the space A^{|ν|+2s} occurs with multiplicity \binom{n+s−2}{n−2}. A highest weight vector Q in A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} of weight −(|ν| + 2s) is the image of the function 1 under an intertwining embedding

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

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

(2)  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}. Let K_{Q} be the embedding defined by K_{Q}(1) = Q. In [R1] it was proved that if

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

then the adjoint map K_{Q}^{∗} is the differential operator given by

(4)  K_{Q}^{∗}(f_{1} ⊗ ⋯ ⊗ f_{n})(z) = ∑_{|t|=s} c̄_{t} \frac{f_{1}^{(t_{1})}(z) ⋯ f_{n}^{(t_{n})}(z)}{(ν_{1})_{t_{1}} ⋯ (ν_{n})_{t_{n}}}.

In particular, writing e_{m}(z) = z^{m},

\begin{aligned}K_{Q}^{∗}(e_{m_{1}} ⊗ ⋯ ⊗ e_{m_{n}}) & = ∑_{|t|=s} c̄_{t} \frac{(−m_{1})_{t_{1}} ⋯ (−m_{n})_{t_{n}}}{(ν_{1})_{t_{1}} ⋯ (ν_{n})_{t_{n}}} (−1)^{s}e_{|m|−s} \\ & = \overline{T Q(m_{1}, …, m_{n})}e_{|m|−s}, \end{aligned}

where T Q is the polynomial of degree s given by

T Q(m) = (−1)^{s} ∑_{|t|=s} c_{t} \frac{(−m_{1})_{t_{1}} ⋯ (−m_{n})_{t_{n}}}{(ν_{1})_{t_{1}} ⋯ (ν_{n})_{t_{n}}} .

This exhibits T Q as a coupling coefficient for our representation. If Q and Q′ are two highest weight vectors, the corresponding coupling coefficients satisfy

(5)  ∑_{|m|=k+s} \frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T Q(m) \overline{T Q′(m)} = \frac{(|ν| + 2s)_{k}}{k!} ⟨Q, Q′⟩,

for k = 0, 1, 2, …. Here s is the degree of Q or Q′; if their degrees are different both sides vanish.

For n = 2, the space of highest weight vectors of degree s is one-dimensional, and spanned by (z_{1} − z_{2})^{s}. The norm of this vector is given by

(6)  ∥(z_{1} − z_{2})^{s}∥^{2} = \frac{s!(ν_{1} + ν_{2} + s − 1)_{s}}{(ν_{1})_{s}(ν_{2})_{s}} .

It will be convenient to write

P_{s}^{(ν_{1},ν_{2})}(m_{1}, m_{2}) = (ν_{1})_{s}(ν_{2})_{s}(T (z_{1} − z_{2})^{s})(m_{1}, m_{2})

= ∑_{k=max(0, s−m_{2})}^{min(s, m_{1})} \binom{s}{k}(−1)^{k} \frac{(ν_{1})_{s}(ν_{2})_{s}}{(ν_{1})_{k}(ν_{2})_{s−k}} (−m_{1})_{k}(−m_{2})_{s−k}. (7)

These polynomials are essentially Clebsch–Gordan coefficients, or Wigner 3j-symbols, of our representation. As is well-known, they can be expressed in terms of Hahn polynomials; in standard notation [KS]

P_{s}^{(ν_{1},ν_{2})}(m_{1}, m_{2}) = (ν_{1})_{s}(−m_{1} − m_{2})_{s} Q_{s}(m_{1}; ν_{1} − 1, ν_{2} − 1, m_{1} + m_{2}).

We will use some well-known properties of Clebsch–Gordan coefficients. First, it is clear from (7) that

(8)  P_{s}^{(ν_{1},ν_{2})}(m_{1}, m_{2}) = \begin{cases}(−1)^{m_{2}}s! \frac{(ν_{1})_{s}(ν_{2})_{s}}{(ν_{1})_{m_{1}}(ν_{2})_{m_{2}}} , & m_{1} + m_{2} = s, \\ 0, & m_{1} + m_{2} < s. \end{cases}

¿From (5) and (6) one obtains the orthogonality relations

(9) ∑_{m_{1}+m_{2}=s+k} \frac{(ν_{1})_{m_{1}}(ν_{2})_{m_{2}}}{m_{1}! m_{2}!} P_{s}^{(ν_{1},ν_{2})}(m_{1}, m_{2})P_{t}^{(ν_{1},ν_{2})}(m_{1}, m_{2})

= \frac{(ν_{1} + ν_{2} + 2s)_{k}}{k!} C_{s}^{ν_{1}ν_{2}} δ_{st},

where

(10)  C_{s}^{ν_{1}ν_{2}} = s!(ν_{1})_{s}(ν_{2})_{s}(ν_{1} + ν_{2} + s − 1)_{s}.

The expression (7) may be rewritten in terms of terminating _{3}F_{2}(1)-series

3F_{2} ( \begin{matrix}−n, a, b \\ c, d \end{matrix} | 1) = ∑_{k=0}^{n} \frac{(−n)_{k}(a)_{k}(b)_{k}}{(c)_{k}(d)_{k}k!} .

Because of the known symmetries of such series, cf. [GR], p. 59, there is a large number of similar expressions, such as

P_{s}^{(ν_{1},ν_{2})}(m_{1}, m_{2})

= ∑_{k=0}^{s} \binom{s}{k}(−1)^{s−k} \frac{(ν_{1})_{s}(ν_{2})_{s}}{(ν_{1})_{k}(ν_{2})_{s−k}} (ν_{1} + m_{1})_{k}(ν_{2} + m_{2})_{s−k} (11)

= (−1)^{s} \frac{m_{2}!}{(m_{1} + m_{2} − s)!}(ν_{1})_{s}(ν_{2} + m_{2})_{m_{1}} _{3}F_{2} ( \begin{matrix}−m_{1}, ν_{1} + s, 1 − ν_{2} − s \\ ν_{1}, 1 − ν_{2} − m_{1} − m_{2} \end{matrix} | 1) (12)

= (ν_{1})_{s}(−m_{1} − m_{2})_{s} _{3}F_{2} ( \begin{matrix}−m_{1}, −s, ν_{1} + ν_{2} + s − 1 \\ ν_{1}, −m_{1} − m_{2} \end{matrix} | 1) . (13)

From the transformation formulas used to derive (12) it follows that

(14)  _{3}F_{2} ( \begin{matrix}−m_{1}, ν_{1} + s, 1 − ν_{2} − s \\ ν_{1}, 1 − ν_{2} − m_{1} − m_{2} \end{matrix} | 1) = 0

if m_{1}, m_{2}, and s are integers with

m_{1}, s ≥ 0, m_{1} + m_{2} ≥ s, m_{2} < 0.

This is a special case of Karlsson’s summation formula [K].

We will now discuss the coupling kernels introduced in [R2]. Let us consider the Hilbert subspace of highest weight vectors of degree s in A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}}. It has a reproducing kernel

K_{s}(z, w̄) = ∑_{|t|=|u|=s} c_{t,u} z_{1}^{t_{1}} ⋯ z_{n}^{t_{n}}w̄_{1}^{u_{1}} ⋯ w̄_{n}^{u_{n}}.

The coupling kernels are obtained by applying the transform T in each of the two variables z and w; thus we define

P_{s}(i, j) = ∑_{|t|=|u|=s} c_{t,u} \frac{(−i)_{t}(−j)_{u}}{(ν)_{t}(ν)_{u}} ,

where we use multi-index notation such as

(−i)_{t} = (−i_{1})_{t_{1}} ⋯ (−i_{n})_{t_{n}}.

This is the reproducing kernel for the space of coupling coefficients of degree s, in the sense that

\frac{(|ν| + 2s)_{k}}{k!} T Q(l) = ∑_{|m|=k+s} \frac{(ν_{1})_{m_{1}} ⋯ (ν_{n})_{m_{n}}}{m_{1}! ⋯ m_{n}!} T Q(m)P_{s}(l, m)

for each k = 0, 1, 2, … and each highest weight vector Q of degree s. If (Q_{k}) and (Q̃_{l}) are any two orthogonal bases of this space, then

(15)  P_{s}(i, j) = ∑_{k, l} \frac{⟨Q_{k}, Q̃_{l}⟩}{∥Q_{k}∥^{2}∥Q̃_{l}∥^{2}} \overline{T Q_{k}(i)} T Q̃_{l}(j).

In [R2] an explicit expression for the coupling kernels was found, namely,

(16) P_{s}(i, j) = \frac{(−1)^{s}(−|i|)_{s}(−|j|)_{s}}{(|ν| + s − 1)_{s}s!} ∑_{m_{1},...,m_{n}=0}^{∞} \frac{(|ν| + s − 1)_{|m|}(−s)_{|m|}(−i)_{m}(−j)_{m}}{(−|i|)_{|m|}(−|j|)_{|m|}m!(ν)_{m}}

(with the convention 0/0 = 0, so that the sum is actually finite). Our proof of the triple sum formula will use this expression, though in fact we only need the case |i| = |j| = s, which is slightly easier to prove.

Let us now see how to introduce recoupling coefficients, and in particular Wigner 9j-symbols, in the present approach. First we define the binary coupling of two highest weight vectors. Suppose that we decompose the tensor product as

A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} ≃ (A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{k}}) ⊗ (A^{ν_{k+1}} ⊗ ⋯ ⊗ A^{ν_{n}}),

where 1 ≤ k ≤ n − 1. Let Q_{1} be a highest weight vector in A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{k}} of weight −μ_{1} and Q_{2} one in A^{ν_{k+1}} ⊗ ⋯ ⊗ A^{ν_{n}} of weight −μ_{2}. Thus

μ_{1} = ν_{1} + ⋯ + ν_{k} + 2t_{1}, μ_{2} = ν_{k+1} + ⋯ + ν_{n} + 2t_{2},

where t_{1} and t_{2} are the degrees of Q_{1} and Q_{2}. For s a non-negative integer we define

(Q_{1}, Q_{2})_{s} = (μ_{1})_{s}(μ_{2})_{s}(K_{Q_{1}} ⊗ K_{Q_{2}})(z_{1} − z_{2})^{s},

which is a highest weight vector in A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} of weight −(μ_{1} + μ_{2} + 2s) and degree t_{1} + t_{2} + s. Equivalently, (Q_{1}, Q_{2})_{s} is determined by

K_{(Q_{1},Q_{2})_{s}}^{∗}(f_{1}⊗⋯⊗f_{n}) = (μ_{1})_{s}(μ_{2})_{s}K_{(z_{1}−z_{2})^{s}}^{∗}(K_{Q_{1}}^{∗}(f_{1}⊗⋯⊗f_{k})⊗K_{Q_{2}}^{∗}(f_{k+1}⊗⋯⊗f_{n})).

Choosing f_{i}(z) = z^{m_{i}} gives the important formula

(17)  T (Q_{1}, Q_{2})_{s}(m) = T Q_{1}(m′) T Q_{2}(m′′) P_{s}^{(μ_{1},μ_{2})}(|m′| − t_{1}, |m′′| − t_{2}),

where

m′ = (m_{1}, …, m_{k}), m′′ = (m_{k+1}, …, m_{n}).

Moreover,

∥(Q_{1}, Q_{2})_{s}∥^{2} = C_{s}^{μ_{1}μ_{2}}∥Q_{1}∥^{2}∥Q_{2}∥^{2},

with the constant as in (10).

We also need a more general type of binary coupling, corresponding to a decomposition

A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} ≃ (A^{ν_{σ_{1}}} ⊗ ⋯ ⊗ A^{ν_{σ_{k}}}) ⊗ (A^{ν_{σ_{k+1}}} ⊗ ⋯ ⊗ A^{ν_{σn}}),

where σ is a permutation of (1, 2, …, n). If Q_{1} and Q_{2} are highest weight vectors in A^{ν_{σ_{1}}} ⊗ ⋯ ⊗ A^{ν_{σ_{k}}} and A^{ν_{σ_{k+1}}} ⊗ ⋯ ⊗ A^{ν_{σn}}, respectively, we define

(Q_{1}, Q_{2})_{s}^{σ}(z_{1}, …, z_{n}) = (Q_{1}, Q_{2})_{s}(z_{σ_{1}}, …, z_{σ_{n}}).

Then (Q_{1}, Q_{2})_{s}^{σ} is a highest weight vector in A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}}.

Now let n = 4 and σ = (1, 3, 2, 4). We consider the two orthogonal systems of highest weight vectors in A^{ν_{1}} ⊗ A^{ν_{2}} ⊗ A^{ν_{3}} ⊗ A^{ν_{4}} defined by

(18)  \begin{aligned}Q_{pqr} & = ((1, 1)_{p}, (1, 1)_{q})_{r}, \\ Q̃_{tuv} & = ((1, 1)_{t}, (1, 1)_{u})_{v}^{σ}. \end{aligned}

Graphically, these iterated couplings may be represented as in Figure 1.

[figure: 1; 2; 3; 4; 1; 3; 2; 4; p; q; t; u; r; v] Figure 1

Let

W_{pqr}^{tuv} = ⟨Q_{pqr}, Q̃_{tuv}⟩,

where the parameters ν_{i} are suppressed from the notation. These are essentially the Wigner 9j-coefficients of our representation. Since W_{pqr}^{tuv} vanishes if p+q+r ≠ t+u+v, we may assume that

p + q + r = t + u + v = s.

In more standard notation for 9j-symbols, we have

\begin{cases}j_{1} & j_{2} & j_{12} \\ j_{3} & j_{4} & j_{34} \\ j_{13} & j_{24} & j \end{cases}\} = \frac{1}{∥Q_{pqr}∥∥Q̃_{tuv}∥} W_{pqr}^{tuv},

where the “angular momenta” are given by

j_{1} = −\frac{1}{2}ν_{1}, j_{2} = −\frac{1}{2}ν_{2}, j_{3} = −\frac{1}{2}ν_{3}, j_{4} = −\frac{1}{2}ν_{4},
j_{12} = −\frac{1}{2}(ν_{1} + ν_{2} + 2p), j_{34} = −\frac{1}{2}(ν_{3} + ν_{4} + 2q),
j_{13} = −\frac{1}{2}(ν_{1} + ν_{3} + 2t), j_{24} = −\frac{1}{2}(ν_{2} + ν_{4} + 2u),
j = −\frac{1}{2}(ν_{1} + ν_{2} + ν_{3} + ν_{4} + 2s),

and the normalizing factor by

∥Q_{pqr}∥∥Q̃_{tuv}∥ = (C_{p}^{ν_{1}ν_{2}}C_{q}^{ν_{3}ν_{4}}C_{r}^{ν_{1}+ν_{2}+2p, ν_{3}+ν_{4}+2q}C_{t}^{ν_{1}ν_{3}}C_{u}^{ν_{2}ν_{4}}C_{v}^{ν_{1}+ν_{3}+2t, ν_{2}+ν_{4}+2u})^{\frac{1}{2}} .

We conclude this section with a remark on transvectants and Wigner symbols. Let us consider the case n = 2 of (4). It will be convenient to write

\begin{aligned}J_{s}^{ν_{1}ν_{2}}(f_{1}, f_{2})(z) & = (ν_{1})_{s}(ν_{2})_{s} K_{(z_{1}−z_{2})^{s}}^{∗}(f_{1} ⊗ f_{2})(z) \\ & = ∑_{k=0}^{s} \binom{s}{k}(−1)^{s−k} \frac{(ν_{1})_{s}(ν_{2})_{s}}{(ν_{1})_{k}(ν_{2})_{s−k}} f_{1}^{(k)}(z)f_{2}^{(s−k)}(z), \end{aligned}

so that, again with e_{m}(z) = z^{m},

J_{s}^{ν_{1}ν_{2}}(e_{m_{1}}, e_{m_{2}}) = P_{s}^{(ν_{1},ν_{2})}(m_{1}, m_{2}) e_{m_{1}+m_{2}−s}.

This operator occurs in classical invariant theory and is known as the transvectant. It has also been called Cohen’s operator and the Rankin–Cohen bracket [Z]. It is clear from the above that the Wigner 9j-symbols occur in the expansion

J_{r}^{ν_{1}+ν_{2}+2p, ν_{3}+ν_{4}+2q}(J_{p}^{ν_{1}ν_{2}}(f_{1}, f_{2}), J_{q}^{ν_{3}ν_{4}}(f_{3}, f_{4}))

= ∑_{t+u+v=p+q+r} \frac{W_{pqr}^{tuv}}{∥Q̃_{tuv}∥^{2}} J_{v}^{ν_{1}+ν_{3}+2t, ν_{2}+ν_{4}+2u}(J_{t}^{ν_{1}ν_{3}}(f_{1}, f_{3}), J_{u}^{ν_{2}ν_{4}}(f_{2}, f_{4})).

It is interesting that in a recent paper [CMZ] the coefficients occurring in

J_{r}^{ν_{1}+ν_{2}+2q, ν_{3}}(J_{q}^{ν_{1}ν_{2}}(f_{1}, f_{2}), f_{3}) = ∑_{t+u=q+r} R_{qr}^{tu} J_{u}^{ν_{1}, ν_{2}+ν_{3}+2t}(f_{1}, J_{t}^{ν_{2}ν_{3}}(f_{2}, f_{3}))

arise in a seemingly different context. These are, up to a normalizing factor, the Racah coefficients, or Wigner 6j-symbols, of our representation.

3. An expansion formula

In this section we will derive a formula expressing the product of one 9j- and two 3j-symbols as a quadruple sum, each term being a product of four 3j-symbols. As a special case we get an expression for the 9j-symbols as a double sum of products of two 3j-symbols, of which the triple sum formula is a rather easy consequence.

Let us choose Q and Q̃ in (15) as the bases given by (18). This gives

(19)  P_{s}(i, j) = ∑_{\begin{matrix}p+q+r=s \\ t+u+v=s \end{matrix}} \frac{W_{pqr}^{tuv}}{∥Q_{pqr}∥^{2}∥Q̃_{tuv}∥^{2}} \overline{T Q_{pqr}(i)} T Q̃_{tuv}(j).

Moreover, by (17),

T Q_{pqr}(i) = P_{p}^{(ν_{1},ν_{2})}(i_{1}, i_{2}) P_{q}^{(ν_{3},ν_{4})}(i_{3}, i_{4}) P_{r}^{(ν_{1}+ν_{2}+2p,ν_{3}+ν_{4}+2q)}(i_{1} + i_{2} − p, i_{3} + i_{4} − q),
T Q̃_{tuv}(j) = P_{t}^{(ν_{1},ν_{3})}(j_{1}, j_{3}) P_{u}^{(ν_{2},ν_{4})}(j_{2}, j_{4}) P_{v}^{(ν_{1}+ν_{3}+2t,ν_{2}+ν_{4}+2u)}(j_{1} + j_{3} − t, j_{2} + j_{4} − u).

Since these are real polynomials we may ignore the conjugate bar in (19). Let us temporarily write P_{p} = P_{p}^{(ν_{1},ν_{2})} and so on. We fix eight non-negative integers

k_{12}, k_{34}, k_{13}, k_{24}, p′, q′, t′, u′,

and apply, to both sides of (19), the summation operator

∑_{\begin{matrix}i_{1}+i_{2}=p′+k_{12} \\ i_{3}+i_{4}=q′+k_{34} \end{matrix}} \frac{(ν)_{i}}{i!} P_{p′}(i_{1}, i_{2})P_{q′}(i_{3}, i_{4}) ∑_{\begin{matrix}j_{1}+j_{3}=t′+k_{13} \\ j_{2}+j_{4}=u′+k_{24} \end{matrix}} \frac{(ν)_{j}}{j!} P_{t′}(j_{1}, j_{3})P_{u′}(j_{2}, j_{4}).

By the orthogonality of Clebsch–Gordan coefficients (9), the right hand side vanishes unless (p′, q′, t′, u′) = (p, q, t, u), and we obtain

(20) ∑_{\begin{matrix}i_{1}+i_{2}=p+k_{12} \\ i_{3}+i_{4}=q+k_{34} \end{matrix}} ∑_{\begin{matrix}j_{1}+j_{3}=t+k_{13} \\ j_{2}+j_{4}=u+k_{24} \end{matrix}} \frac{(ν)_{i}(ν)_{j}}{i! j!} P_{s}(i, j)P_{p}(i_{1}, i_{2})P_{q}(i_{3}, i_{4})P_{t}(j_{1}, j_{3})P_{u}(j_{2}, j_{4})

\begin{matrix}= \frac{(ν_{1} + ν_{2} + 2p)_{k_{12}}(ν_{3} + ν_{4} + 2q)_{k_{34}}(ν_{1} + ν_{3} + 2t)_{k_{13}}(ν_{2} + ν_{4} + 2u)_{k_{24}}}{k_{12}! k_{34}! k_{13}! k_{24}! C_{r}^{ν_{1}+ν_{2}+2p, ν_{3}+ν_{4}+2q}C_{v}^{ν_{1}+ν_{3}+2t, ν_{2}+ν_{4}+2u}} \\ × W_{pqr}^{tuv} P_{r}(k_{12}, k_{34})P_{v}(k_{13}, k_{24}). \end{matrix}

Now we expand P_{s} according to (16). Then the left hand side of (20) takes the form

(−1)^{s} \frac{(−p − q − k_{12} − k_{34})_{s}(−t − u − k_{13} − k_{24})_{s}}{(|ν| + s − 1)_{s}s!} (21)

× ∑_{m_{1},m_{2},m_{3},m_{4}=0}^{∞} \frac{(|ν| + s − 1)_{|m|}(−s)_{|m|}}{(−p − q − k_{12} − k_{34})_{|m|}(−t − u − k_{13} − k_{24})_{|m|}m!(ν)_{m}}

× ∑_{i_{1}+i_{2}=p+k_{12}} \frac{(ν_{1})_{i_{1}}(ν_{2})_{i_{2}}}{i_{1}! i_{2}!} (−i_{1})_{m_{1}}(−i_{2})_{m_{2}}P_{p}(i_{1}, i_{2})

× ∑_{i_{3}+i_{4}=q+k_{34}} \frac{(ν_{3})_{i_{3}}(ν_{4})_{i_{4}}}{i_{3}! i_{4}!} (−i_{3})_{m_{3}}(−i_{4})_{m_{4}}P_{q}(i_{3}, i_{4})

× ∑_{j_{1}+j_{3}=t+k_{13}} \frac{(ν_{1})_{j_{1}}(ν_{3})_{j_{3}}}{j_{1}! j_{3}!} (−j_{1})_{m_{1}}(−j_{3})_{m_{3}}P_{t}(j_{1}, j_{3})

× ∑_{j_{2}+j_{4}=u+k_{24}} \frac{(ν_{2})_{j_{2}}(ν_{4})_{j_{4}}}{j_{2}! j_{4}!} (−j_{2})_{m_{2}}(−j_{4})_{m_{4}}P_{u}(j_{2}, j_{4}).

The four inner sums may be computed using the probably well-known formula

(22) ∑_{i_{1}+i_{2}=s+k} \frac{(ν_{1})_{i_{1}}(ν_{2})_{i_{2}}}{i_{1}! i_{2}!} (−i_{1})_{m_{1}}(−i_{2})_{m_{2}}P_{s}^{(ν_{1},ν_{2})}(i_{1}, i_{2})

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

Note that, by (8), the left hand side vanishes unless m_{1} + m_{2} ≤ s + k, and the right hand side unless m_{1} + m_{2} ≥ s. One may prove (22) by inserting expression (11) for the Clebsch–Gordan coefficients (see the proof of Lemma 5.1 below).

It follows that the expression (21) equals

(−1)^{s} \frac{(−p − q − k_{12} − k_{34})_{s}(−t − u − k_{13} − k_{24})_{s}}{(|ν| + s − 1)_{s}s!}

× ∑_{m_{1},m_{2},m_{3},m_{4}=0}^{∞} \frac{(|ν| + s − 1)_{|m|}(−s)_{|m|}(ν)_{m}}{(−p − q − k_{12} − k_{34})_{|m|}(−t − u − k_{13} − k_{24})_{|m|}m!}

× \frac{(ν_{1} + ν_{2} + p + m_{1} + m_{2})_{k_{12}+p−m_{1}−m_{2}}(ν_{3} + ν_{4} + q + m_{3} + m_{4})_{k_{34}+q−m_{3}−m_{4}}}{(k_{12} + p − m_{1} − m_{2})!(k_{34} + q − m_{3} − m_{4})!}

× \frac{(ν_{1} + ν_{3} + t + m_{1} + m_{3})_{k_{13}+t−m_{1}−m_{3}}(ν_{2} + ν_{4} + u + m_{2} + m_{4})_{k_{24}+u−m_{2}−m_{4}}}{(k_{13} + t − m_{1} − m_{3})!(k_{24} + u − m_{2} − m_{4})!}

× P_{p}(m_{1}, m_{2})P_{q}(m_{3}, m_{4})P_{t}(m_{1}, m_{3})P_{u}(m_{2}, m_{4}).

On the other hand, this is equal to the right hand side of (20). After some cancellations, this gives the following formula.

Theorem 3.1. Let k_{12}, k_{34}, k_{13}, k_{24}, p, q, r, t, u, and v be non-negative integers with p + q + r = t + u + v = s. Then, in the notation above, the equality

\frac{1}{k_{12}! k_{34}! k_{13}! k_{24}! C_{r}^{ν_{1}+ν_{2}+2p, ν_{3}+ν_{4}+2q}C_{v}^{ν_{1}+ν_{3}+2t, ν_{2}+ν_{4}+2u}}

× W_{pqr}^{tuv} P_{r}^{(ν_{1}+ν_{2}+2p,ν_{3}+ν_{4}+2q)}(k_{12}, k_{34})P_{v}^{(ν_{1}+ν_{3}+2t,ν_{2}+ν_{4}+2u)}(k_{13}, k_{24})

= (−1)^{s} \frac{(−p − q − k_{12} − k_{34})_{s}(−t − u − k_{13} − k_{24})_{s}}{(|ν| + s − 1)_{s}s!}

× ∑_{m_{1},m_{2},m_{3},m_{4}} \frac{(|ν| + s − 1)_{|m|}(−s)_{|m|}(ν)_{m}}{(−p − q − k_{12} − k_{34})_{|m|}(−t − u − k_{13} − k_{24})_{|m|}m!}

× \frac{1}{(k_{12}+p−m_{1}−m_{2})!(k_{34}+q−m_{3}−m_{4})!(k_{13}+t−m_{1}−m_{3})!(k_{24}+u−m_{2}−m_{4})!}

× \frac{1}{(ν_{1}+ν_{2}+2p)_{m_{1}+m_{2}−p}(ν_{3}+ν_{4}+2q)_{m_{3}+m_{4}−q}(ν_{1}+ν_{3}+2t)_{m_{1}+m_{3}−t}(ν_{2}+ν_{4}+2u)_{m_{2}+m_{4}−u}}

× P_{p}^{(ν_{1},ν_{2})}(m_{1}, m_{2})P_{q}^{(ν_{3},ν_{4})}(m_{3}, m_{4})P_{t}^{(ν_{1},ν_{3})}(m_{1}, m_{3})P_{u}^{(ν_{2},ν_{4})}(m_{2}, m_{4})

holds, where the sum is over all non-negative integers such that

\begin{cases}p ≤ m_{1} + m_{2} ≤ p + k_{12}, \\ q ≤ m_{3} + m_{4} ≤ q + k_{34}, \\ t ≤ m_{1} + m_{3} ≤ t + k_{13}, \\ u ≤ m_{2} + m_{4} ≤ u + k_{24}, \\ m_{1} + m_{2} + m_{3} + m_{4} ≤ s. \end{cases}

To prove the triple sum formula we only need the case

k_{12} = k_{13} = 0, k_{34} = r, k_{24} = v.

Then the two conditions

m_{1} + m_{2} = p, m_{1} + m_{3} = t

hold, so that the sum in the theorem reduces to a double sum. Moreover, the factors P_{r}, P_{v}, P_{p}, and P_{t} become trivial by (8), so we obtain a formula expressing the Wigner 9j-symbol as a double sum of products of two Clebsch–Gordan coefficients. After further simplifications, such as writing

k_{34} + q − m_{3} − m_{4} = s − |m|,

we obtain the following formula.

Corollary 3.2. For p, q, r, t, u, and v non-negative integers with

p + q + r = t + u + v = s,

WIGNER 9j-SYMBOLS 11

the equality

W_{pqr}^{tuv} = (−1)^{r+v+s} \frac{p! r! t! v!(ν_{1})_{p}(ν_{2})_{p}(ν_{1})_{t}(ν_{3})_{t}}{(|ν| + s − 1)_{s}}

× (|ν| + 2p + 2q + r − 1)_{r}(|ν| + 2t + 2u + v − 1)_{v}

× ∑_{m_{1},m_{2},m_{3},m_{4}} \frac{(|ν| + s − 1)_{|m|}(ν_{4})_{m_{4}}}{m!(s − |m|)!(ν_{1})_{m_{1}}} (−1)^{m_{1}+m_{4}}P_{q}^{(ν_{3},ν_{4})}(m_{3}, m_{4})P_{u}^{(ν_{2},ν_{4})}(m_{2}, m_{4})

× (ν_{2} + ν_{4} + u + m_{2} + m_{4})_{s−|m|}(ν_{3} + ν_{4} + q + m_{3} + m_{4})_{s−|m|}

holds, where the sum is over all non-negative integers such that

\begin{cases}m_{1} + m_{2} = p, \\ q ≤ m_{3} + m_{4} ≤ q + r, \\ m_{1} + m_{3} = t, \\ u ≤ m_{2} + m_{4} ≤ u + v. \end{cases}

4. The triple sum formula

We will now prove the triple sum formula. Let us rewrite the sum in Corollary 3.2 using j = s − |m| and m_{1} as summation variables, thus replacing m_{2}, m_{3}, and m_{4} by

p − m_{1}, t − m_{1}, s + m_{1} − p − t − j,

respectively. Using the identities

(a)_{s−k} = (−1)^{k} \frac{(a)_{s}}{(1 − a − s)_{k}} ,

(a)_{k} = (−1)^{k}(1 − a − k)_{k},

we obtain

W_{pqr}^{tuv} = (−1)^{q+u}r! v!(|ν| + 2p + 2q + r − 1)_{r}(|ν| + 2t + 2u + v − 1)_{v}

× (ν_{1})_{p}(ν_{2})_{p}(ν_{1})_{t}(ν_{3})_{t} ∑_{j,m_{1}} \frac{(1 − ν_{2} − ν_{4} − 2u − v)_{j}(1 − ν_{3} − ν_{4} − 2q − r)_{j}}{j!(2 − |ν| − 2s)_{j}}

× \frac{(−p)_{m_{1}}(−t)_{m_{1}}}{m_{1}!(ν_{1})_{m_{1}}} \frac{(ν_{4})_{s+m_{1}−p−t−j}}{(s + m_{1} − p − t − j)!}

× P_{q}^{(ν_{3},ν_{4})}(t − m_{1}, s + m_{1} − p − t − j) P_{u}^{(ν_{2},ν_{4})}(p − m_{1}, s + m_{1} − p − t − j),

where the sum is over j and m_{1} such that

\begin{cases}0 ≤ m_{1} ≤ min(p, t), \\ 0 ≤ j ≤ min(r, v), \\ p + t + j ≤ s + m_{1}. \end{cases}

Now let us expand P_{q} and P_{u} using formulas (12) and (13), respectively. Thus we write

P_{q} = (−1)^{q} \frac{(s + m_{1} − p − t − j)!}{(r − j)!}(ν_{3})_{q}(ν_{4} + s + m_{1} − p − t − j)_{t−m_{1}}

× ∑_{i=0}^{t−m_{1}} \frac{(m_{1} − t)_{i}(ν_{3} + q)_{i}(1 − ν_{4} − q)_{i}}{i!(ν_{3})_{i}(1 − ν_{4} − q − r + j)_{i}} ,

P_{u} = (ν_{2})_{u}(j − u − v)_{u} ∑_{k=0}^{min(u, p−m_{1})} \frac{(−u)_{k}(m_{1} − p)_{k}(ν_{2} + ν_{4} + u − 1)_{k}}{k!(j − u − v)_{k}(ν_{2})_{k}} .

We then change the order of summation, pulling m_{1} inside. We also manipulate the Pochhammer symbols to make i, j, k appear only as subscripts. This gives

W_{pqr}^{tuv} = (u + v)!(ν_{1})_{p}(ν_{1})_{t}(ν_{2})_{p}(ν_{2})_{u}(ν_{3})_{q}(ν_{3})_{t}(ν_{4})_{q+r}

× (|ν| + 2p + 2q + r − 1)_{r}(|ν| + 2t + 2u + v − 1)_{v}

× ∑_{i=0}^{t} ∑_{j=0}^{min(r, v)} ∑_{k=0}^{min(p, u)} \frac{1}{i! j! k!} \frac{1}{(1 − ν_{4} − q − r)_{i+j}(−u − v)_{j+k}}

× \frac{(−r)_{j}(−v)_{j}(1 − ν_{3} − ν_{4} − 2q − r)_{j}(1 − ν_{2} − ν_{4} − 2u − v)_{j}}{(2 − |ν| − 2s)_{j}}

× \frac{(−t)_{i}(ν_{3} + q)_{i}(1 − ν_{4} − q)_{i}}{(ν_{3})_{i}} \frac{(−p)_{k}(−u)_{k}(ν_{2} + ν_{4} + u − 1)_{k}}{(ν_{2})_{k}}

× ∑_{m_{1}=max(0, p+t+j−s)}^{min(t−i, p−k)} \frac{(i − t)_{m_{1}}(k − p)_{m_{1}}}{m_{1}!(ν_{1})_{m_{1}}} .

However, it follows from (14), that if we instead first sum over i, then the sums for which m_{1} ≤ p + t + j − s (if there are any such) vanish. Thus we can replace the lower limit in the inner sum by 0, and then compute it using the Chu–Vandermonde formula:

∑_{m_{1}} \frac{(i − t)_{m_{1}}(k − p)_{m_{1}}}{m_{1}!(ν_{1})_{m_{1}}} = \frac{(ν_{1})_{p+t−i−k}}{(ν_{1})_{t−i}(ν_{1})_{p−k}} = \frac{(ν_{1})_{p+t}}{(ν_{1})_{p}(ν_{1})_{t}} \frac{(1 − ν_{1} − t)_{i}(1 − ν_{1} − p)_{k}}{(1 − ν_{1} − p − t)_{i+k}} .

This gives the following result, which is equivalent to the Ališauskas–Jucys triple sum formula. As remarked in [SR2], the sum may be expressed in terms of generalized hypergeometric functions introduced by Srivastava [S].

Theorem 4.1. The Wigner 9j-symbols for 𝔰𝔲(1, 1) are given by

W_{pqr}^{tuv} = (u + v)!(ν_{1})_{p+t}(ν_{2})_{p}(ν_{2})_{u}(ν_{3})_{q}(ν_{3})_{t}(ν_{4})_{q+r} × (|ν| + 2p + 2q + r − 1)_{r}(|ν| + 2t + 2u + v − 1)_{v}

× ∑_{i=0}^{t} ∑_{j=0}^{min(r, v)} ∑_{k=0}^{min(p, u)} \frac{1}{i! j! k!} \frac{1}{(1 − ν_{4} − q − r)_{i+j}(1 − ν_{1} − p − t)_{i+k}(−u − v)_{j+k}}

× \frac{(−t)_{i}(1 − ν_{1} − t)_{i}(ν_{3} + q)_{i}(1 − ν_{4} − q)_{i}}{(ν_{3})_{i}}

× \frac{(−r)_{j}(−v)_{j}(1 − ν_{3} − ν_{4} − 2q − r)_{j}(1 − ν_{2} − ν_{4} − 2u − v)_{j}}{(2 − |ν| − 2s)_{j}}

× \frac{(−p)_{k}(−u)_{k}(1 − ν_{1} − p)_{k}(ν_{2} + ν_{4} + u − 1)_{k}}{(ν_{2})_{k}} ,

where p + q + r = t + u + v = s.

This formula might look awkward, but we stress that it is the most compact known expression for Wigner 9j-symbols. Note that the parameter occurring in the Pochhammer symbol ( )_{i+j} may be written in two ways:

1 − ν_{4} − q − r = (1 − ν_{4} − q) + (−r) = (ν_{3} + q) + (1 − ν_{3} − ν_{4} − 2q − r),

as a sum of those occurring in ( )_{i} and ( )_{j} in the numerator. This is also true for ( )_{i+k} and ( )_{j+k}, and each of the twelve numerator parameters occur once in the corresponding six sums.

5. General recoupling coefficients

In this section we will extend Theorem 3.1 to arbitrary recoupling coefficients. Thus we consider two decompositions:

A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}} ≃ (A^{ν_{σ_{1}}} ⊗ ⋯ ⊗ A^{ν_{σ_{l}}}) ⊗ (A^{ν_{σ_{l+1}}} ⊗ ⋯ ⊗ A^{ν_{σn}}) = B_{1} ⊗ B_{2}
≃ (A^{ν_{τ_{1}}} ⊗ ⋯ ⊗ A^{ν_{τ_{l′}}}) ⊗ (A^{ν_{τ_{l′+1}}} ⊗ ⋯ ⊗ A^{ν_{τn}}) = B′_{1} ⊗ B′_{2},

where σ and τ are permutations and 1 ≤ l, l′ ≤ n − 1. We choose arbitrary highest weight vectors

Q_{1} ∈ B_{1}, Q_{2} ∈ B_{2}, Q′_{1} ∈ B′_{1}, Q′_{2} ∈ B′_{2},

of respective degree

s_{1}, s_{2}, s′_{1}, s′_{2},

and weight

−μ_{1}, −μ_{2}, −μ′_{1}, −μ′_{2}.

For a multi-index (i_{1}, …, i_{n}) we write

i_{1}^{σ} = (i_{σ_{1}}, …, i_{σ_{l}}), i_{2}^{σ} = (i_{σ_{l+1}}, …, i_{σ_{n}}),
i_{1}^{τ} = (i_{τ_{1}}, …, i_{τ_{l′}}), i_{2}^{τ} = (i_{τ_{l′+1}}, …, i_{τ_{n}}),

so that

μ_{1} = |ν_{1}^{σ}| + 2s_{1}, μ_{2} = |ν_{2}^{σ}| + 2s_{2},
μ′_{1} = |ν_{1}^{τ}| + 2s′_{1}, μ′_{2} = |ν_{2}^{τ}| + 2s′_{2}.

We will study the recoupling coefficient

⟨(Q_{1}, Q_{2})_{s_{3}}^{σ}, (Q′_{1}, Q′_{2})_{s′_{3}}^{τ}⟩,

where we may assume that

s_{1} + s_{2} + s_{3} = s′_{1} + s′_{2} + s′_{3} = s.

In (19) we replace Q_{pqr} and Q̃_{tuv} by suitable orthogonal systems of highest weight vectors containing (Q_{1}, Q_{2})_{s_{3}}^{σ} and (Q′_{1}, Q′_{2})_{s′_{3}}^{τ}. Using (5) instead of (9) we obtain

∑_{\begin{matrix}|i_{1}^{σ}|=s_{1}+k_{1}, |i_{2}^{σ}|=s_{2}+k_{2} \\ |j_{1}^{τ} |=s′_{1}+k′_{1}, |j_{2}^{τ} |=s′_{2}+k′_{2} \end{matrix}} \frac{(ν)_{i}(ν)_{j}}{i! j!} P_{s}(i, j) T Q_{1}(i_{1}^{σ}) T Q_{2}(i_{2}^{σ}) \overline{T Q′_{1}(j_{1}^{τ}) T Q′_{2}(j_{2}^{τ})}

= \frac{(μ_{1})_{k_{1}}(μ_{2})_{k_{2}}(μ′_{1})_{k′_{1}}(μ′_{2})_{k′_{2}}}{k_{1}! k_{2}! k′_{1}! k′_{2}! C_{s_{3}}^{μ_{1}μ_{2}}C_{s′_{3}}^{μ′_{1}μ′_{2}}} ⟨(Q_{1}, Q_{2})_{s_{3}}^{σ}, (Q′_{1}, Q′_{2})_{s′_{3}}^{τ}⟩ P_{s_{3}}^{(μ_{1},μ_{2})}(k_{1}, k_{2})P_{s′_{3}}^{(μ′_{1},μ′_{2})}(k′_{1}, k′_{2}),

where k_{1}, k_{2}, k′_{1}, k′_{2} are arbitrary non-negative integers. By (16) the left hand side equals

(−1)^{s} \frac{(−s_{1} − s_{2} − k_{1} − k_{2})_{s}(−s′_{1} − s′_{2} − k′_{1} − k′_{2})_{s}}{(|ν| + s − 1)_{s}s!}

× ∑_{m} \frac{(|ν| + s − 1)_{|m|}(−s)_{|m|}}{(−s_{1} − s_{2} − k_{1} − k_{2})_{|m|}(−s′_{1} − s′_{2} − k′_{1} − k′_{2})_{|m|}m!(ν)_{m}}

× ∑_{|i_{1}^{σ}|=s_{1}+k_{1}} \frac{(ν_{1}^{σ})_{i_{1}^{σ}}}{i_{1}^{σ}!} (−i_{1}^{σ})_{m_{1}^{σ}} T Q_{1}(i_{1}^{σ}) ∑_{|i_{2}^{σ}|=s_{2}+k_{2}} \frac{(ν_{2}^{σ})_{i_{2}^{σ}}}{i_{2}^{σ}!} (−i_{2}^{σ})_{m_{2}^{σ}} T Q_{2}(i_{2}^{σ})

× ∑_{|j_{1}^{τ} |=s′_{1}+k′_{1}} \frac{(ν_{1}^{τ})_{j_{1}^{τ}}}{j_{1}^{τ}!} (−j_{1}^{τ})_{m_{1}^{τ}} \overline{T Q′_{1}(j_{1}^{τ})} ∑_{|j_{2}^{τ} |=s′_{2}+k′_{2}} \frac{(ν_{2}^{τ})_{j_{2}^{τ}}}{j_{2}^{τ}!} (−j_{2}^{τ})_{m_{2}^{τ}} \overline{T Q′_{2}(j_{2}^{τ})}.

To compute the inner sums we need instead of (22) the following lemma.

Lemma 5.1. If Q is a highest weight vector of degree s in A^{ν_{1}} ⊗ ⋯ ⊗ A^{ν_{n}}, then

∑_{|i|=s+k} \frac{(ν)_{i}}{i!} (−i)_{m} T Q(i) = (−1)^{|m|} \frac{(|ν| + s + |m|)_{k+s−|m|}}{(k + s − |m|)!} (ν)_{m} T Q(m).

In [R2] this is derived as a limit case of a more general formula. We will give an independent proof here.

Proof. Let Q be given by (3). Then, by (2),

\begin{aligned}∑_{|t|=s} c_{t} z^{t} & = ∑_{|t|=s} c_{t}(1 + z_{1})^{t_{1}} ⋯ (1 + z_{n})^{t_{n}} \\ & = ∑_{|t|=s} c_{t} ∑_{k_{1},...,k_{n}≥0} \frac{(−t)_{k}}{k!} (−1)^{|k|}z^{k}. \end{aligned}

Applying T on both sides gives

(23)  T Q(m) = ∑_{|t|=s} c_{t} ∑_{k_{1},...,k_{n}≥0} \frac{(−t)_{k}(−m)_{k}}{k!(ν)_{k}} = ∑_{|t|=s} c_{t} \frac{(ν + m)_{t}}{(ν)_{t}} ,

by the Chu–Vandermonde formula; that is T Q(m) = (−1)^{s}T Q(−ν − m). This generalizes expression (11). Thus we have

\begin{aligned}∑_{|i|=s+k} \frac{(ν)_{i}}{i!} (−i)_{m} T Q(i) & = ∑_{|i|=s+k} \frac{(ν)_{i}}{i!} (−i)_{m} ∑_{|t|=s} c_{t} \frac{(ν + i)_{t}}{(ν)_{t}} \\ & = ∑_{|t|=s} c_{t} ∑_{|i|=s+k} \frac{(ν + t)_{i}}{i!} (−i)_{m}. \end{aligned}

Replacing i by i + m in the inner sum gives

∑_{|i|=s+k} \frac{(ν + t)_{i}}{i!} (−i)_{m} = ∑_{|i|=s+k−|m|} \frac{(ν + t)_{i+m}}{(i + m)!} (−i − m)_{m}

= (−1)^{|m|}(ν +t)_{m} ∑_{|i|=s+k−|m|} \frac{(ν + t + m)_{i}}{i!} = (−1)^{|m|}(ν +t)_{m} \frac{(|ν| + |t| + |m|)_{s+k−|m|}}{(s + k − |m|)!} ,

where the last step is the multivariable Chu–Vandermonde formula. This gives

∑_{|i|=s+k} \frac{(ν)_{i}}{i!} (−i)_{m} T Q(i) = (−1)^{|m|}(ν)_{m} \frac{(|ν| + s + |m|)_{s+k−|m|}}{(s + k − |m|)!} ∑_{|t|=s} c_{t} \frac{(ν + m)_{t}}{(ν)_{t}} ,

which, again by (23), proves the lemma. □

Applying this lemma gives the following generalization of Theorem 3.1.

Theorem 5.2. In the notation above, the equality

\frac{1}{k_{1}! k_{2}! k′_{1}! k′_{2}! C_{s_{3}}^{μ_{1}μ_{2}}C_{s′_{3}}^{μ′_{1}μ′_{2}}} ⟨(Q_{1}, Q_{2})_{s_{3}}^{σ}, (Q′_{1}, Q′_{2})_{s′_{3}}^{τ}⟩ P_{s_{3}}^{(μ_{1},μ_{2})}(k_{1}, k_{2})P_{s′_{3}}^{(μ′_{1},μ′_{2})}(k′_{1}, k′_{2})

= (−1)^{s} \frac{(−s_{1} − s_{2} − k_{1} − k_{2})_{s}(−s′_{1} − s′_{2} − k′_{1} − k′_{2})_{s}}{(|ν| + s − 1)_{s}s!}

× ∑_{m_{1},...,m_{n}} \frac{(|ν| + s − 1)_{|m|}(−s)_{|m|}(ν)_{m}}{(−s_{1} − s_{2} − k_{1} − k_{2})_{|m|}(−s′_{1} − s′_{2} − k′_{1} − k′_{2})_{|m|}m!}

× \frac{1}{(k_{1} + s_{1} − |m_{1}^{σ}|)!(k_{2} + s_{2} − |m_{2}^{σ}|)!(k′_{1} + s′_{1} − |m_{1}^{τ}|)!(k′_{2} + s′_{2} − |m_{2}^{τ}|)!}

× \frac{1}{(μ_{1})_{|m_{1}^{σ}|−s_{1}}(μ_{2})_{|m_{2}^{σ}|−s_{2}}(μ′_{1})_{|m_{1}^{τ}|−s′_{1}}(μ′_{2})_{|m_{2}^{τ}|−s′_{2}}}

× T Q_{1}(m_{1}^{σ}) T Q_{2}(m_{2}^{σ}) \overline{T Q′_{1}(m_{1}^{τ}) T Q′_{2}(m_{2}^{τ})}

holds, where the sum is over all non-negative integers such that

\begin{cases}s_{1} ≤ |m_{1}^{σ}| ≤ s_{1} + k_{1}, \\ s_{2} ≤ |m_{2}^{σ}| ≤ s_{2} + k_{2}, \\ s′_{1} ≤ |m_{1}^{τ}| ≤ s′_{1} + k′_{1}, \\ s′_{2} ≤ |m_{2}^{τ}| ≤ s′_{2} + k′_{2}. \end{cases}

We remark that for Wigner 6j-symbols, that is, for the recoupling coefficient

⟨((1, 1)_{q}, 1)_{r}, (1, (1, 1)_{t})_{u}⟩,

then in certain cases such as

k_{1} = k′_{2} = 0, k_{2} = r, k′_{1} = u,

the sum in the theorem reduces to a single hypergeometric sum, which is in fact a terminating balanced _{4}F_{3}(1)-series. This gives a new proof of Racah’s single sum formula for 6j-symbols.

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