AVERAGE NORMS OF POLYNOMIALS

Toufik Mansour ^{1}

Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden

                              toufik@math.chalmers.se

Abstract

In this paper we study the average L_{2α}-norm over T -polynomials, where α is a positive integer. More precisely, we present an explicit formula for the average L_{2α}-norm over all the polynomials of degree exactly n with coefficients in T, where T is a finite set of complex numbers, α is a positive integer, and n ≥ 0. In particular, we give a complete answer for the cases of Littlewood polynomials and polynomials of a given height. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.

2000 Mathematics Subject Classification: 33C50, 42C05

1. Introduction

The set of polynomials with special coefficients has given much attention and there are many old research questions concerning it. Erdös and Szekere; Hilbert; Littlewood; Prouhet, Tarry, and Escott, are suggested number of these questions (see [B3, Pages 5-7]). For example, Prouhet, Tarry, and Escott asked to find a polynomial with integer coefficients that is divisible by (z −1)^{n} and has smallest sum of the absolute values of the coefficients, and Erdös and Szekere asked to find the minimum of || ∏_{j=1}^{n}(1 − z^{a_{j}})||_{∞} where the a_{j} are positive integers, for given n. The problem to find the maximum and the minimum norms of polynomials with special coefficients is a specific old and difficult problem. In the L_{4}-norm this problem is often called Golay’s ”Merit Factor” problem. In the supremum norm this problem is due to Littlewood. These problems are at least fifty years old and never had solved. In this paper we interested in the following problem (for particular cases see [B3, page 35]): Find the average L_{2α}-norm over polynomials of degree exactly n with coefficients in a given finite set T, where α is a positive integer. In this paper, we give a complete answer for this problem.

Let T = {x_{1}, x_{2}, …, x_{d}} be any finite set of complex numbers. A polynomial p(z) = a_{n}z^{n}+⋯+a_{1}z+a_{0} is said to be T -polynomial if a_{i} ∈ T for all i, 0 ≤ i ≤ n. We denote by 𝔗_{T} (n) the set of all T -polynomials of degree exactly n. For example, if T = {0, 1, 2} then the set of T -polynomials of degree exactly 1 is given by 𝔗_{T} (1) = {z, 2z, z + 1, 2z + 1, z + 2, 2z + 2}. The cardinality of this set, 𝔗_{T} (n), is denoted by N_{T} (n). Clearly, for all n ≥ 0,

N_{T} (n) = \{ \begin{matrix}d^{n+1}, & 0 ∉ T and n ≥ 1, \\ (d − 1)d^{n}, & 0 ∈ T and n ≥ 1, \\ d, & n = 0. \end{matrix}

1Research financed by EC’s IHRP Programme, within the Research Training Network ”Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272

A polynomial p(z) is said to be Littlewood polynomial if it {−1, 1}-polynomial. A polynomial p(z) is said to be polynomial of height h if it {−h, −h + 1, …, h − 1, h}-polynomial. For example, p(z) = z^{2} − z + 1 is a Littlewood polynomial of degree 2, and p(z) = z^{3} − 2z^{2} − 1 is a polynomial of height 2 and degree 3.

Let p(z) be any T -polynomial of degree exactly n. For any positive integer α, the L_{α}-norm on the boundary of the unit disk is defined by

||p||_{α} = (\frac{1}{2π} ∫_{0}^{2π} |p(e^{iθ})|^{α} dθ)^{\frac{1}{α}} .

Let f, g, h be any three ℝ-polynomials, the (f, g, h)-average over T -polynomials of degree exactly n is defined by

(1)  E_{T} (n; f, g, h) = \frac{1}{2πN_{T} (n)} ∑_{p∈𝔗_{T} (n)} ∫_{0}^{2π} h(e^{iθ})f(p(e^{iθ}))g(p(e^{−iθ})) dθ,

for any n ≥ 1. We denote by e_{T} (n, s, t, m) the (z^{s}, z^{t}, z^{m})-average over T -polynomials of degree exactly n, where m ∈ ℤ and s, t, n ≥ 0.

We define the average L_{α}-norm over T -polynomials of degree exactly n by

(2)  μ_{T}^{α}(n) = e_{T} (n; α/2, α/2, 0) = \frac{1}{N_{T} (n)} ∑_{p∈𝔗_{T} (n)} ||p||_{α}^{α} = \frac{1}{2πN_{T} (n)} ∑_{p∈𝔗_{n}} ∫_{0}^{2π} |p(e^{iθ})|^{α} dθ,

for any positive integer α. For α = 0 we define μ_{T}^{0} (n) = 1 for all n ≥ 1.

Some one could ask to find an explicit formula for μ_{T}^{2α}(n), where T is a finite set of complex numbers and α is a positive integer (for particular cases, see the research problem in [B3, Page 35]). While the cases of Littlewood polynomials and polynomials of height h have attracted much attention (for example, see [B3, BC, NB]), the case of other sets T. The case of T = {−1, 1}, Littlewood polynomials, considered by several authors as follows. In 1990, Newman and Byrnes [NB] found μ_{{−1,1}}^{4}(n) = 2n^{2} + 3n + 1. In 2002, Borwein and Choi [BC] proved

\begin{matrix}μ_{\{−1,1\}}^{6}(n) = 6n^{3} + 9n^{2} + 4n + 1, \\ μ_{\{−1,1\}}^{8}(n) = 24n^{4} + 30n^{3} + 4n^{2} + 5n + 4 − 3(−1)^{n}. \end{matrix}

In the case T = \{−1, 0, 1\}, polynomials of height 1, Borwein [B2] proved μ_{\{−1,0,1\}}^{2}(n) = \frac{2}{3}(n + 1), μ_{\{−1,0,1\}}^{4}(n) = \frac{2}{9}(4n^{2}+7n+3), and μ_{\{−1,0,1\}}^{6}(n) = \frac{2}{9}(8n^{3}+18n^{2}+13n+3). More generally, Borwein [B2] found for any h ≥ 0,

\begin{matrix}μ_{\{−h,−h+1,...,h−1,h\}}^{2}(n) = \frac{h(h+1)}{3}(n + 1), \\ μ_{\{−h,−h+1,...,h−1,h\}}^{4}(n) = \frac{h(h+1)}{45}(10h(h + 1)n^{2} + (19h^{2} + 19h − 3)n + 3(3h^{2} + 3h − 1)). \end{matrix}

In this paper we suggest a general approach to study the average L_{2α}-norm over T -polynomials of degree exactly n, for any positive integer α and any finite set T of complex numbers, which allows one to get an explicit expression for μ_{T}^{2α}(n). More precisely, we find an explicit expression for the generating function e_{T} (x, u, v, w). Using this generating function we get an explicit expression for e_{T} (n; s, t, m) in general, and μ_{T}^{2α}(n) = e_{T} (n, α, α, 0) in particular, where m ∈ ℤ, s, t, n ≥ 0, α is a positive integer, and T is a finite set of complex numbers. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.

The main result of this paper can be formulated as follows. We denote by e_{T} (x, u, v, w) the generating function for the sequence {e_{T} (n, s, t, m)}_{n,s,t,m}, that is,

e_{T} (x, u, v, w) = ∑_{m∈ℤ} ∑_{p,q,n≥0} e_{T} (n, s, t, m)x^{n}u^{s}v^{t}w^{m}.

Theorem 1.1. Let T = {x_{1}, x_{2}, …, x_{d}} be a finite set of complex numbers. Then the generating function e_{T} (x, u, v, w) is given by

∑_{n≥1} [\frac{1}{d^{n}} ∑_{j_{1},...,j_{n}=1}^{d} \frac{x^{n−1}}{(1 − x_{j_{1}}u − x_{j_{2}}uw^{−1} − ⋯ − x_{j_{n}}uw^{−n+1}) (1 − \overline{x_{j_{1}}}v − \overline{x_{j_{2}}}vw − ⋯ − \overline{x_{j_{n}}}vw^{n−1})}] .

Moreover, e_{T} (n, s, t, m) is given by

\frac{1}{d^{n+1}} ∑_{j_{1},...,j_{n+1}=1}^{d} ∑_{s,t≥0} ∑_{\begin{matrix}k_{1} + ⋯ + k_{n+1} = s \\ ℓ_{1} + ⋯ + ℓ_{n+1} = t \\ ∑_{a=1}^{n+1}(a − 1)(ℓ_{a} − k_{a}) = m \end{matrix}} ∏_{a=1}^{n+1}(x_{j_{a}}^{t_{a}} \overline{x_{j_{a}}}^{r_{a}})\binom{s}{k_{1}, …, k_{n+1}}\binom{t}{ℓ_{1}, …, ℓ_{n+1}}.

Using Theorem 1.1 we get an explicit expression for the average L_{2α}-norm over T -polynomials of degree exactly n, namely μ_{T}^{2α}(n). Corollary 1.2. Let T = {x_{1}, x_{2}, …, x_{d}} be a finite set of complex numbers. Then the average L_{2α}- norm over T -polynomials of degree exactly n, namely μ_{T}^{2α}(n), is given by

\frac{1}{d^{n+1}} ∑_{j_{1},...,j_{n+1}=1}^{d} ∑_{\begin{matrix}k_{1} + ⋯ + k_{n+1} = α \\ ℓ_{1} + ⋯ + ℓ_{n+1} = α \\ ∑_{a=1}^{n+1}(a − 1)(ℓ_{a} − k_{a}) = 0 \end{matrix}} ∏_{a=1}^{n+1}(x_{j_{a}}^{k_{a}} \overline{x_{j_{a}}}^{ℓ_{a}})\binom{α}{k_{1}, …, k_{n+1}}\binom{α}{ℓ_{1}, …, ℓ_{n+1}}.

Proof. If applying Theorem 1.1 and the identity e_{T} (n, α, α, 0) = μ_{T}^{2α}(n), then we get the desired result. □

The paper is organized as follows. The proof of our main result, Theorem 1.1, is presented in Section 2. In section 3 we present a general application for our results. In particular, we give an explicit formulas up to α = 4 where ∑_{t∈T} t = 0 (as in the case of Littlewood polynomials and polynomials of height h). In Section 4 we apply our main result for ceratin sets T, which allows us to get more details in the cases of Littlewood polynomials and polynomials of height 1. Finally, in Section 5 we suggest several directions to generalize the results of the previous sections.

2. Proofs

Let us start by introduce the quantity that plays the crucial role in the proof of Theorem 1.1. Theorem 2.1. Let T = {x_{1}, x_{2}, …, x_{d}} be a set of complex numbers, n ≥ 1, m any integer, and s, t ≥ 0. Then

(3)  e_{T} (n, s, t, m) = \frac{1}{d} ∑_{j=1}^{d} ∑_{k=0}^{a} ∑_{ℓ=0}^{b} x_{j}^{s−k} \overline{x_{j}} ^{t−ℓ}\binom{s}{k}\binom{t}{ℓ}e_{T} (n − 1, k, ℓ, m + k − ℓ).

Proof. Let z = e^{iθ} and f_{s,t,m}(p(z)) = z^{m}p^{s}(z)\overline{p}^{t}(z), where \overline{p}(z) is the conjugate polynomial of the polynomial p(z) ∈ 𝔗_{T} (n). Since for any T -polynomial p(z) ∈ 𝔗_{T} (n) there exists an unique polynomial q(z) ∈ 𝔗_{T} (n − 1) and exists j, 1 ≤ j ≤ d, such that p(z) = zq(z) + x_{j}, we have that

∑_{p(z)∈𝔗_{T} (n)} f_{s,t,m}(p(z)) = z^{m} ∑_{j=1}^{d} ∑_{q(z)∈𝔗_{T} (n−1)}(zq(z) + x_{j})^{s}(\overline{z} \overline{q}(z) + \overline{x_{j}})^{t}.

Using (x + y)^{k} = ∑_{j=0}^{k} \binom{k}{j}x^{j}y^{k−j} and \overline{z} = z^{−1} we get

∑_{p(z)∈𝔗_{T} (n)} f_{s,t,m}(p(z)) = ∑_{q(z)∈𝔗_{T} (n−1)} ∑_{j=1}^{d} ∑_{k=0}^{s} ∑_{ℓ=0}^{t} x_{j}^{s−k} \overline{x_{j}} ^{t−ℓ}\binom{s}{k}\binom{t}{ℓ}f_{k,ℓ,m+k−ℓ}(q(z)).

Therefore, using Definition 1 we get the desired result. □

To present Recurrence 3 in terms of generating functions we need the following lemma. Lemma 2.2. Let F (x, y) = ∑_{s,t≥0} d_{s,t}x^{s}y^{t} be a generating function with two variables. Then

∑_{s,t≥0} x^{s}y^{t} (∑_{k=0}^{s} ∑_{ℓ=0}^{t} a^{s−k}b^{t−ℓ}d_{k,ℓ}\binom{s}{k}\binom{t}{ℓ}) = \frac{1}{(1 − ax)(1 − by)}F (\frac{x}{1 − ax}, \frac{y}{1 − by} ) .

Proof. By definitions we have

\frac{1}{(1 − ax)(1 − by)}F (\frac{x}{1 − ax}, \frac{y}{1 − by} ) = ∑_{s,t≥0} a^{−s}b^{−t}d_{s,t} \frac{(ax)^{s}(by)^{t}}{(1 − ax)^{s+1}(1 − by)^{t+1}} .

Using \frac{x^{r}}{(1−x)^{r}} = ∑_{n≥0} \binom{n}{r}x^{r} we get

\frac{1}{(1 − ax)(1 − by)}F (\frac{x}{1 − ax}, \frac{y}{1 − by} ) = ∑_{s,t≥0} ( ∑_{k,ℓ≥0} a^{k−s}b^{ℓ−t}d_{s,t}\binom{k}{s}\binom{ℓ}{t}x^{k}y^{ℓ}) ,

equivalently,

\frac{1}{(1 − ax)(1 − by)}F (\frac{x}{1 − ax}, \frac{y}{1 − by} ) = ∑_{s,t≥0} x^{s}y^{t} (∑_{k=0}^{s} ∑_{ℓ=0}^{t} a^{s−k}b^{t−ℓ}d_{k,ℓ}\binom{s}{k}\binom{t}{ℓ}) ,

as claimed. □

Remark 2.3. Lemma 2.2 can be generalized as follows. Let

F (v_{1}, …, v_{k}) = ∑_{s_{1},s_{2},...,s_{k}≥0} e_{s_{1},...,s_{k}} ∏_{d=1}^{k} v_{d}^{s_{d}}

be a generating function with k variables. Then

∑_{s_{1},s_{2},...,s_{k}≥0} ∏_{d=1}^{k} v_{d}^{s_{d}} ( ∑_{j_{1}=0}^{s_{1}} ∑_{j_{2}=0}^{s_{2}} ⋯ ∑_{j_{k}=0}^{s_{k}} e_{j_{1},j_{2},...,j_{k}} ∏_{d=1}^{k} \binom{s_{d}}{j_{d}}w_{d}^{s_{d}−j_{d}} )

is given by

\frac{1}{∏_{d=1}^{k}(1 − v_{d}w_{d})}F (\frac{v_{1}}{1 − w_{1}v_{1}} , \frac{v_{2}}{1 − w_{2}v_{2}} , …, \frac{v_{k}}{1 − w_{k}v_{k}} ) .

Now we are ready to prove our main result, namely Theorem 1.1. Theorem 2.4. Let T = {x_{1}, x_{2}, …, x_{d}} be a finite set of complex numbers. Then the generating function e_{T} (x, v, u, w) is given by

∑_{n≥1} [\frac{1}{d^{n}} ∑_{j_{1},...,j_{n}=1}^{d} \frac{x^{n−1}}{(1 − x_{j_{1}}u − x_{j_{2}}uw^{−1} − ⋯ − x_{j_{n}}uw^{−n+1}) (1 − \overline{x_{j_{1}}}v − \overline{x_{j_{2}}}vw − ⋯ − \overline{x_{j_{n}}}vw^{n−1})}] .

Proof. If multiplying Equation 3 by x^{n}u^{s}v^{t}w^{m}, and summing over all m ∈ ℤ, s, t ≥ 0, and n ≥ 1, together with using Lemma 2.2, then we arrive to

e_{T} (x, u, v, w) − ∑_{m∈ℤ} ∑_{s,t≥0} e(0, s, t, m)u^{s}v^{t}w^{m} = \frac{x}{d} ∑_{j=1}^{d} [\frac{1}{(1−x_{j}u)(1−\overline{x_{j}}v)} e_{T} (x, \frac{u}{w(1−x_{j}u)}, \frac{wv}{1−\overline{x_{j}}v} , w)] .

On the other hand, by definitions we have that e_{T} (0, s, t, m) = \frac{1}{d} ∑_{j=1}^{d} δ_{m}x_{j}^{s}\overline{x_{j}}^{t} for any m ∈ ℤ and s, t ≥ 0, where δ_{m} = 1 if m = 0, otherwise δ_{m} = 0. So

∑_{m∈ℤ} ∑_{s,t≥0} e_{T} (0, s, t, m)u^{s}v^{t}w^{m} = \frac{1}{d} ∑_{j=1}^{d} \frac{1}{(1 − x_{j}u)(1 − \overline{x_{j}}v)}.

Therefore, by combining the above two equations we get that

(4)  e_{T} (x, u, v, w) = \frac{1}{d} ∑_{j=1}^{d} \frac{1}{(1 − x_{j}u)(1 − \overline{x_{j}}v)} [1 + xe_{T} (x, \frac{u}{w(1 − x_{j}u)}, \frac{wv}{1 − x_{j}v} , w)] .

An infinite number of applications of this identity concludes the proof. □

Remark 2.5. Theorem 2.4 yields the generating function e_{T} (x, u, v, w) is symmetric under the translation (u, v, w) → (v, u, w^{−1}), that is, e_{T} (x, u, v, w) = e_{T} (x, v, u, w^{−1}).

Theorem 2.4 can be presented as follows. Corollary 2.6. Let T = {x_{1}, x_{2}, …, x_{d}} be a finite set of complex numbers. Then the generating function e_{T} (x, v, u, w) is given by

∑_{n≥1} ∑_{j_{1},...,j_{n}=1}^{d} ∑_{s,t≥0} ∑_{\begin{matrix}k_{1} + ⋯ + k_{n} = s \\ ℓ_{1} + ⋯ + ℓ_{n} = t \end{matrix}} ∏_{a=1}^{n}(x_{j_{a}}^{t_{a}} \overline{x_{j_{a}}}^{r_{a}})\binom{s}{k_{1}, …, k_{n}}\binom{t}{ℓ_{1}, …, ℓ_{n}} \frac{x^{n−1}u^{s}v^{t}w^{( ∑_{a=1}^{n}(a−1)(r_{a}−t_{a}))}}{d^{n}} .

Proof. Using Theorem 2.4 we get that the generating function e_{T} (x, v, u, w) is given by

∑_{n≥1} [ ∑_{j_{1},j_{2},...,j_{n}=1}^{d} ∑_{s,t≥0} (x_{j_{1}} + x_{j_{2}}w^{−1} + ⋯ + x_{j_{n}}w^{−n+1})^{s} (\overline{x_{j_{1}}} + \overline{x_{j_{2}}}w + ⋯ + \overline{x_{j_{n}}}w^{n−1})^{t} \frac{x^{n−1}u^{s}v^{t}}{d^{n}} ] .

the rest is easy to check by the identity (a_{1} + … + a_{n})^{s} = ∑_{k_{1}+···+k_{n}=s} \binom{s}{k_{1},...,k_{n}} ∏_{j=1}^{n} a_{j}^{k_{j}}. □

Let us denote by μ_{T}^{α}(x) the generating function for the sequence {μ_{T}^{α}(n)}_{n≥0}, that is, μ_{T}^{α}(x) = ∑_{n≥0} μ_{T}^{α}(n)x^{n}. Corollary 1.2 gives a complete answer to find the generating function μ_{T}^{2α}(x) for any given finite set T and positive integer α.

Example 2.7. Using Corollary 1.2 for α = 1 we get that

μ_{T}^{2} (x) = ∑_{n≥1} ∑_{j_{1},...,j_{n}=1}^{d} (∑_{k=1}^{n} |x_{j_{k}}|^{2}) \frac{x^{n−1}}{d^{n}} = ∑_{n≥1} (nd^{n−1} ∑_{j=1}^{d} |x_{j}|^{2}) \frac{x^{n−1}}{d^{n}} ,

so it is easy to see that μ_{T}^{2} (x) = \frac{1}{d(1−x)} ∑_{k=1}^{n} |x_{j_{k}}|^{2} + xμ_{T}^{2} (x), hence

μ_{T}^{2} (x) = ∑_{n≥0} (\frac{n + 1}{d} ∑_{j=1}^{d} |x_{j}|^{2}) x^{n}.

In particular, we have that μ_{\{−1,1\}}(n; 2) = n + 1, and μ_{\{−1,0,1\}}(n; 2) = \frac{2}{3}(n + 1).

Corollary 1.2 provides a finite algorithm for finding the average μ_{T}^{2α}(n) where n, T, and α are given. This algorithm has been implemented in MAPLE, and yields explicit results for given n, T, and α (see the tables below).

Remark 2.8. To apply Corollary 1.2 we have to consider d^{n} possibilities for j_{i} and \binom{n+α−1}{α}^{2} possibilities for k_{i} and ℓ_{i}, namely we have to consider d^{n}\binom{n+α−1}{α}^{2} possibilities. Therefore, Corollary 1.2, for n large, works very slowly.

3. Exact formulas

Corollary 1.2 provides a close formula for finding the average L_{2α}-norm over T -polynomials of degree exactly n for any given n ≥ 0 and α ≥ 1. Remark 2.8 yields that the problem to find exact formula for μ_{T}^{2α}(n) with given only α ≥ 1 it is a hard problem by using Corollary 1.2. Thus, we suggest here another approach to find an explicit formula for μ_{T}^{2α}(n).

First of all, let us denote by e_{T}^{s,t}(x, w) the (s + t)-derivative of the generating function e_{T} (x, u, v, w) with respect u^{s} and then with respect v^{t} at u = v = 0, that is,

e_{T}^{s,t} = e_{T}^{s,t}(x, w) = \frac{∂^{s+t}}{∂u^{s}∂v^{t}} e_{T} (x, u, v, w)|_{u=v=0} .

For any s, t ≥ 0, we define

A_{T}^{s,t} = ∑_{j=1}^{d} x_{j}^{s}\overline{x_{j}}^{t}.

Now let us consider Equation 4. This equation provide a finite algorithm, μ-algorithm, for finding e_{T} (n, s, t, m) in general, and μ_{T}^{2α}(n) in particular, since s! t! e_{T} (n, s, t, m) is the coefficient of w^{m}x^{n} in the (s+t)-derivative of the generating function e_{T} (x, u, v, w) with respect to u^{s} and then with respect v^{t} at u = v = 0, namely e_{T}^{s,t}(x, w), and μ_{T}^{2α}(n) = e_{T} (n; α, α, 0). Therefore, the μ-algorithm with input α and output μ_{T}^{2α}(x) can be constructed as follows:

(1) Apply the derivative operator with respect u^{s} and then with respect v^{t} on Equation 4 for all s, t, where 0 ≤ s, t ≤ α.

(2) Find explicitly e_{T}^{s,t} for all s, t, where 0 ≤ s, t ≤ α. This by solving the system equations which is obtained from step 1.

(3) Find μ_{T}^{2α}(x), which is the free coefficient of w in e_{T}^{α,α}(x, w).

This algorithm has been implemented in MAPLE , and yields explicit results for given α. Below we present several explicit calculations.

3.1. Formula for μ_{T}^{2} (n). Let us start by apply the μ-algorithm for α = 1. The first step of the μ-algorithm gives

\{ \begin{matrix}e_{T}^{0,0} = 1 + xe_{T}^{0,0}, \\ e_{T}^{0,1} = \frac{1}{d}A_{T}^{0,1} + \frac{x}{d}A_{T}^{0,1}e_{T}^{0,0} + \frac{x}{w} e_{T}^{0,1}, \\ e_{T}^{1,0} = \frac{1}{d}A_{T}^{1,0} + \frac{x}{d}A_{T}^{1,0}e_{T}^{0,0} + \frac{x}{w} e_{T}^{1,0}, \\ e_{T}^{1,1} = \frac{1}{d}A_{T}^{1,1} + \frac{x}{d}A_{T}^{1,1}e_{T}^{0,0} + \frac{xw}{d} A_{T}^{1,0}e_{T}^{0,1} + \frac{x}{dw} A_{T}^{0,1}e_{T}^{1,0} + xe_{T}^{1,1}. \end{matrix}

Equivalently (the second step of the μ-algorithm),

e_{T}^{0,0} = \frac{1}{1 − x}, e_{T}^{1,0} = \frac{A_{T}^{1,0}}{d(1 − x)(1 − xw^{−1})}, e_{T}^{0,1} = \frac{A_{t}^{0,1}}{d(1 − x)(1 − xw)},

and

e_{T}^{1,1} = \frac{1}{d(1−x)^{2}} A_{T}^{1,1} + (\frac{xw}{d^{2}(1−x)^{2}(1−xw)} + \frac{xw^{−1}}{d^{2}(1−x)^{2}(1−xw^{−1})}) A_{T}^{1,0}A_{T}^{0,1}.

Therefore, the third step of the μ-algorithm gives μ_{T}^{2} (x), which is the free coefficient of w in e_{T}^{1,1}(x, w). Hence, we get the following result. Corollary 3.1. We have

μ_{T}^{2} (x) = \frac{1}{d(1 − x)^{2}} A_{T}^{1,1} = \frac{1}{d(1 − x)^{2}} ∑_{j=1}^{d} |x_{j}|^{2}.

Moreover, for all n ≥ 0,

μ_{T}^{2} (n) = \frac{n + 1}{d}A_{T}^{1,1} = \frac{n + 1}{d} ∑_{j=1}^{d} |x_{j}|^{2}.

For example, in the case of Littlewood polynomials, namely T = {−1, 1}, we have that μ_{T}^{2} (n) = n+1, and in the case of polynomials of height h, namely T = {−h, −h + 1, …, h − 1, h}, we have that μ_{T}^{2} (n) = \frac{h(h+1)}{3}(n + 1).

3.2. Formula for μ_{T}^{4} (n). Again, using the μ-algorithm for α = 2 we get that

e_{T}^{0,2} = \frac{2}{d(1−x)(1−xw^{2})}A_{T}^{0,2} + \frac{2xw}{d(1−xw^{2})}e_{T}^{0,1}A_{T}^{0,1},
e_{T}^{2,0} = \frac{2}{d(1−x)(1−xw^{−2})}A_{T}^{2,0} + \frac{2x}{dw(1−xw^{−2})}e_{T}^{1,0}A_{T}^{1,0},

\begin{aligned}e_{T}^{1,2} & = \frac{2}{d(1−x)(1−xw)}A_{T}^{1,2} + \frac{4xw}{d(1−xw)}e_{T}^{0,1}A_{T}^{1,1} + \frac{xw^{2}}{d(1−xw)}e_{T}^{0,2}A_{T}^{1,0} + \frac{2x}{dw(1−xw)}e_{T}^{1,0}A_{T}^{0,2} \\ & +\frac{4x}{d(1−xw)}e_{T}^{1,1}A_{T}^{0,1}, \end{aligned}

\begin{aligned}e_{T}^{2,1} & = \frac{2}{d(1−x)(1−xw^{−1})}A_{T}^{2,0} + \frac{4x}{dw(1−xw^{−1})}e_{T}^{1,0}(x, w)A_{T}^{1,1} + \frac{x}{dw^{2}(1−xw^{−1})}e_{T}^{0,2}A_{T}^{0,1} + \frac{2xw}{d(1−xw^{−1})}e_{T}^{1,0}A_{T}^{2,0} \\ & +\frac{4x}{d(1−xw^{−1})}e_{T}^{1,1}A_{T}^{1,0}, \end{aligned}

\begin{aligned}e_{T}^{2,2} & = 2\frac{4}{d(1−x)^{2}} A_{T}^{2,2} + \frac{8xw}{d(1−x)}e_{T}^{0,1}A_{T}^{2,1} + \frac{2xw^{2}}{d(1−x)}e_{T}^{0,2}A_{T}^{2,0} + \frac{2x}{dw^{2}(1−x)}e_{T}^{2,0}A_{T}^{0,2} + \frac{8x}{dw(1−x)}e_{T}^{1,0}A_{T}^{1,2} \\ & +\frac{16x}{d(1−x)}e_{T}^{1,1}A_{T}^{1,1} + \frac{4xw}{d(1−x)}e_{T}^{1,2}A_{T}^{1,0} + \frac{4x}{dw(1−x)}e_{T}^{2,1}A_{T}^{0,1} \end{aligned}

Solving this linear system in e_{T}^{s,t} where 0 ≤ s, t ≤ 2, we get an explicit expression for e_{T}^{2,2}(x, w) (it is long to present here). Using this expression we find the free coefficient w of e_{T}^{2,2}(x, w), hence we get the following result. Corollary 3.2. We have

\begin{aligned}μ_{T}^{4} (x) & = \frac{1}{d(1−x)^{2}} A_{T}^{2,2} + \frac{4x}{d^{2}(1−x)^{3}} (A_{T}^{1,1})^{2} + \frac{2x^{2}(1+x)^{2}}{d^{3}(1−x^{2})^{3}} [(A_{T}^{1,0})^{2} A_{T}^{0,1} + (A_{T}^{0,1})^{2} A_{T}^{1,0}] \\ & +\frac{8x^{3}}{d^{4}(1−x)^{4}(1+x)} (A_{T}^{1,0})^{2} (A_{T}^{0,1})^{2} . \end{aligned}

Moreover, for all n ≥ 0,

μ_{T}^{4} (n) = ^{\frac{2(A_{T}^{1,0})^{2}(A_{T}^{0,1})^{2}}{3d^{4}}} n^{3} + \{^{\frac{2(A_{T}^{1,1})^{2}}{d^{2}}} + ^{\frac{[(A_{T}^{1,0})^{2}A_{T}^{0,1}+(A_{T}^{0,1})^{2}A_{T}^{1,0}]}{2d^{3}}} − ^{\frac{(A_{T}^{1,0})^{2}(A_{T}^{0,1})^{2}}{d^{4}}} \} n^{2}
+ \{^{\frac{2(A_{T}^{1,1})^{2}}{d^{2}}} + \frac{A_{T}^{2,2}}{d} − ^{\frac{2(A_{T}^{1,0})^{2}(A_{T}^{0,1})^{2}}{3d^{4}}} \} n + \frac{A_{T}^{2,2}}{d} + ^{\frac{(1−(−1)^{n})(A_{T}^{1,0})^{2}(A_{T}^{0,1})^{2}}{2d^{4}}}
−^{\frac{(1−(−1)^{n})[(A_{T}^{1,0})^{2}A_{T}^{0,1}+(A_{T}^{0,1})^{2}A_{T}^{1,0}]}{3d^{3}}}.

For example, in the case of Littlewood polynomials we have that μ_{{−1,1}}^{4}(n) = 2n^{2} + 3n + 1, and in the case of polynomials of height h we get that

μ_{\{−h,−h+1,...,h−1,h\}}^{4}(n; 4) = \frac{h(h + 1)}{45}(10h(h + 1)n^{2} + (19h^{2} + 19h − 3)n + 3(3h^{2} + 3h − 1)).

3.3. Formula for μ_{T}^{2α}(n) where A_{T}^{1,0} = 0. Similarly to the previous subsection, our results can be extended to the case of μ_{T}^{6} (n). Since the answers become very cumbersome, we present here only the simplest case when A_{T}^{1,0} = ∑_{j=1}^{d} x_{j} = 0. Therefore, if we apply our approach for finding μ_{T}^{2α}(n) where A_{T}^{1,0} = 0 (so A_{T}^{0,1} = 0), then we get the following result.

Corollary 3.3. Let T = {x_{1}, x_{2}, …, x_{d}} such that ∑_{j=1}^{d} x_{j} = 0. Then

(i)

μ_{T}^{2} (x) = \frac{1}{d(1 − x)^{2}} A_{T}^{1,1}.

(ii)

μ_{T}^{4} (x) = \frac{1}{d(1 − x)^{2}} A_{T}^{2,2} + \frac{4x}{d^{2}(1 − x)^{3}} (A_{T}^{1,1})^{2} .

(iii)

μ_{T}^{6} (x) = \frac{1}{d(1 − x)^{2}} A_{T}^{3,3} + \frac{18x}{d^{2}(1 − x)^{3}} A_{T}^{1,1}A_{T}^{2,2} + \frac{36x^{2}}{d^{3}(1 − x)^{4}} (A_{T}^{1,1})^{3}.

(iv)

μ_{t}^{8}(x) = \frac{1}{d(1−x)^{2}} A_{T}^{4,4} + \frac{32x}{d^{2}(1−x)^{3}} A_{T}^{1,1}A_{T}^{3,3} + \frac{36x^{2}}{d^{2}(1−x)^{4}} (A_{T}^{2,2})^{2} + \frac{432x^{2}}{d^{3}(1−x)^{4}} A_{T}^{2,2}(A_{T}^{1,1})^{2}
+\frac{72x^{4}(3−2x−2x^{2}+3x^{3}−x^{4})}{d^{4}(1−x)^{4}(1+x)}(A_{T}^{0,2})^{2}(A_{T}^{2,0})^{2} + \frac{576x^{3}}{d^{4}(1−x)^{5}} (A_{T}^{1,1})^{4}
+\frac{48x^{3}}{d^{3}(1−x)^{2}(1−x^{3})}(A_{T}^{0,3}A_{T}^{2,0}A_{T}^{2,1} + A_{T}^{3,0}A_{T}^{0,2}A_{T}^{1,2})
+\frac{72x^{2}}{d^{3}(1−x)^{2}(1−x^{2})}((A_{T}^{1,2})^{2}A_{T}^{2,0} + (A_{T}^{2,1})^{2}A_{T}^{0,2})
+\frac{6x^{2}}{d^{3}(1−x)^{2}(1−x^{2})}((A_{T}^{2,0})^{2}A_{T}^{0,4} + (A_{T}^{0,2})^{2}A_{T}^{4,0}).

4. Applications

In this section we discus particular cases of Theorem 1.1 and Corollary 1.2. More precisely, we apply Corollary 1.2 for ceratin finite sets of complex numbers (special real numbers), and then we present explicit formulas for μ_{T}^{2α} for certain positive integers α.

4.1. Littlewood polynomials. Here we suggest formulas for μ_{T}^{2α}(n), where T = {−1, 1} (Littlewood polynomials). Corollary 1.2 for T = {−1, 1} gives the following result. Corollary 4.1. The average L_{2α}-norm over Littlewood polynomials of degree exactly n, namely μ_{{−1,1}}^{2α}(n), is given by

\frac{1}{2^{n+1}} ∑_{j_{1},...,j_{n+1}=1}^{2} ∑_{\begin{matrix}k_{1} + ⋯ + k_{n+1} = α \\ ℓ_{1} + ⋯ + ℓ_{n+1} = α \\ ∑_{s=1}^{n+1}(s − 1)(ℓ_{s} − k_{s}) = 0 \end{matrix}} (−1)^{∑_{s=1}^{n+1} j_{s}(k_{s}+ℓ_{s})}\binom{α}{k_{1}, …, k_{n+1}}\binom{α}{ℓ_{1}, …, ℓ_{n+1}}.

Using Lemma 4.1 we quickly generate the numbers μ_{{−1,1}}^{2α}(n); the first few of these numbers are given in Table 1.

\begin{matrix}n\α & 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 2 & 6 & 20 & 70 & 252 \\ 2 & 1 & 3 & 15 & 93 & 651 & 4913 \\ 3 & 1 & 4 & 28 & 256 & 2812 & 35024 \\ 4 & 1 & 5 & 45 & 545 & 8149 & 143945 \\ 5 & 1 & 6 & 66 & 996 & 18882 & 433116 \\ 6 & 1 & 7 & 81 & 1645 & 37759 & 1062697 \\ 7 & 1 & 8 & 120 & 2528 & 68152 & 2272128 \\ 8 & 1 & 9 & 153 & 3681 & 113961 & 4385969 \\ 9 & 1 & 10 & 190 & 5140 & 179710 & 7839260 \\ 10 & 1 & 11 & 231 & 6941 & 270451 & 13178561 \end{matrix}

Table 1. Values of μ_{{−1,1}}^{2α}(n).

On the other hand, if applying Equation 4 for T = {−1, 1} then we get that

(5)  \begin{aligned}e_{\{−1,1\}}(x, u, v, w) & = \frac{1}{2(1−u)(1−v)} [1 + xe_{\{−1,1\}} (x, \frac{u}{w(1−u)}, \frac{wv}{1−v} , w)] \\ & +\frac{1}{2(1+u)(1+v)} [1 + xe_{\{−1,1\}} (x, \frac{u}{w(1+u)}, \frac{wv}{1+v} , w)] . \end{aligned}

Now, if applying μ-algorithm on Equation 5 for T = {−1, 1} and α = 0, 1, …, 5, then we get the following result. Corollary 4.2. For all n ≥ 0,

(i) μ_{{−1,1}}^{0}(n) = 1,

(ii) μ_{{−1,1}}^{2}(n) = n + 1,

10 AVERAGE NORMS OF POLYNOMIALS

(iii) μ_{{−1,1}}^{4}(n) = 2n^{2} + 3n + 1, (iv) μ_{{−1,1}}^{6}(n) = 6n^{3} + 9n^{2} + 4n + 1,

(v) μ_{{−1,1}}^{8}(n) = 24n^{4} + 30n^{3} + 4n^{2} + 5n + 4 − 3(−1)^{n}, (vi) μ_{{−1,1}}^{10}(n) = 120n^{5} + 150n^{4} − 350n^{3} + 265n^{2} + 281n − 144 − 5(−1)^{n}(15n − 29). 4.2. Polynomials of height 1. Here we suggest, in the case of polynomials of height 1, some formulas for μ_{{−1,0,1}}^{2α}(n). Using Lemma 4.1 we quickly generate the numbers μ_{{−1,0,1}}^{2α}(n); the first few of these numbers are given in Table 2.

\begin{matrix}n\α & 0 & 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & \frac{2}{3} & \frac{6}{9} & \frac{6}{9} & \frac{6}{9} & \frac{18}{27} \\ 1 & 1 & \frac{4}{3} & \frac{28}{9} & \frac{84}{9} & \frac{284}{9} & \frac{3036}{27} \\ 2 & 1 & \frac{6}{3} & \frac{66}{9} & \frac{330}{9} & \frac{2018}{9} & \frac{42334}{27} \\ 3 & 1 & \frac{8}{3} & \frac{120}{9} & \frac{840}{9} & \frac{7480}{9} & \frac{239832}{27} \\ 4 & 1 & \frac{10}{3} & \frac{190}{9} & \frac{1710}{9} & \frac{19902}{9} & \frac{856010}{27} \\ 5 & 1 & \frac{12}{3} & \frac{276}{9} & \frac{3036}{9} & \frac{43604}{9} & \frac{2348788}{27} \\ 6 & 1 & \frac{14}{3} & \frac{378}{9} & \frac{4914}{9} & \frac{83866}{9} & \frac{5410646}{27} \\ 7 & 1 & \frac{16}{3} & \frac{496}{9} & \frac{7440}{9} & \frac{147056}{9} & \frac{11040304}{27} \\ 8 & 1 & \frac{18}{3} & \frac{630}{9} & \frac{10710}{9} & \frac{240502}{9} & \frac{20567042}{27} \\ 9 & 1 & \frac{20}{3} & \frac{780}{9} & \frac{14820}{9} & \frac{372620}{9} & \frac{35735180}{27} \\ 10 & 1 & \frac{22}{3} & \frac{946}{9} & \frac{19866}{9} & \frac{552786}{9} & \frac{58715598}{27} \end{matrix}

Table 2. Values of μ_{{−1,0,1}}^{2α}(n).

Applying Equation 4 for T = {−1, 0, 1} then we get that

(6)  \begin{aligned}e_{\{−1,0,1\}}(x, u, v, w) & = \frac{1}{3} [1 + xe_{\{−1,0,1\}} (x, \frac{u}{w} , wv, w)] \\ & +\frac{1}{3(1−u)(1−v)} [1 + xe_{\{−1,0,1\}} (x, \frac{u}{w(1−u)}, \frac{wv}{1−v} , w)] \\ & +\frac{1}{3(1+u)(1+v)} [1 + xe_{\{−1,0,1\}} (x, \frac{u}{w(1+u)}, \frac{wv}{1+v} , w)] . \end{aligned}

Now, by applying μ-algorithm on Equation 6 for T = {−1, 0, 1} and α = 0, 1, …, 5 we get as follows. Corollary 4.3. For all n ≥ 0,

(i) μ_{{−1,0,1}}^{0}(n) = 1, (ii) μ_{\{−1,0,1\}}^{2}(n) = \frac{2}{3}(n + 1), (iii) μ_{n,\{−1,0,1\}}^{4}(n) = \frac{2}{9}(4n^{2} + 7n + 3), (iv) μ_{\{−1,0,1\}}^{6}(n) = \frac{2}{9}(8n^{3} + 18n^{2} + 13n + 3),

(v) μ_{\{−1,0,1\}}^{8}(n) = \frac{2}{27}(64n^{4} + 176n^{3} + 128n^{2} + 37n + 15 − 6(−1)^{n}), (vi) μ_{\{−1,0,1\}}^{10}(n) = \frac{2}{81}(640n^{5} + 2400n^{4} + 630n^{2} + 1337n − 363 − 30(−1)^{n}(10n − 13).

4.3. Polynomials with coefficients 0, 1. Here we suggest another case, the case of polynomials with coefficients 0, 1, which allows us to find explicit formulas for μ_{{0,1}}^{2α}(n). Using Lemma 4.1 we quickly generate the numbers μ_{{0,1}}^{2α}(n); the first few of these numbers are given in Table 3.

\begin{matrix}n\α & 0 & 1 & 2 & 3 \\ 0 & 1 & \frac{1}{2} & \frac{1}{2} & \frac{4}{8} \\ 1 & 1 & \frac{2}{2} & \frac{4}{2} & \frac{44}{8} \\ 2 & 1 & \frac{3}{2} & \frac{10}{2} & \frac{204}{8} \\ 3 & 1 & \frac{4}{2} & \frac{19}{2} & \frac{592}{8} \\ 4 & 1 & \frac{5}{2} & \frac{32}{2} & \frac{1397}{8} \\ 5 & 1 & \frac{6}{2} & \frac{49}{2} & \frac{2826}{8} \\ 6 & 1 & \frac{7}{2} & \frac{71}{2} & \frac{5206}{8} \\ 7 & 1 & \frac{8}{2} & \frac{98}{2} & \frac{8876}{8} \\ 8 & 1 & \frac{9}{2} & \frac{131}{2} & \frac{14334}{8} \\ 9 & 1 & \frac{10}{2} & \frac{170}{2} & \frac{22084}{8} \\ 10 & 1 & \frac{11}{2} & \frac{216}{2} & \frac{32828}{8} \end{matrix}

Table 3. Values of μ_{{0,1}}^{2α}(n).

Applying Equation 4 for T = {0, 1} then we get that (7)

e_{\{0,1\}}(x, u, v, w) = \frac{1}{2} [1 + xe_{\{0,1\}} (x, \frac{u}{w} , wv, w)] + \frac{1}{2(1−u)(1−v)} [1 + xe_{\{0,1\}} (x, \frac{u}{w(1−u)}, \frac{wv}{1−v} , w)] .

Now, if applying μ-algorithm on Equation 7 for T = {0, 1} and α = 0, 1, …, 5, then we get the following result. Corollary 4.4. For all n ≥ 0,

(i) μ_{{0,1}}^{0}(n) = 1,

(ii) μ_{\{0,1\}}^{2}(n) = \frac{1}{2}(n + 1),

(iii) μ_{\{0,1\}}^{4}(n) = \frac{1}{96}(4n^{3} + 54n^{2} + 92n + 45 + 3(−1)^{n}),

(iv) μ_{\{0,1\}}^{6}(n) = \frac{1}{2560}(22n^{5} + 460n^{4} + 3100n^{3} + 5600n^{2} + 4143n + 1130 + 75(−1)^{n}(3n + 2)).

4.4. Example of polynomials with coefficients 0, i where i^{2} = −1. Here we extend our examples (see the Section 1) to the case of sets of complex numbers. For example, if we apply the main results of the pervious sections for T = {0, i} then we get the following result. Corollary 4.5. For all n ≥ 0,

(i) μ_{{0,i}}^{0}(n) = 1,

(ii) μ_{\{0,i\}}^{2}(n) = \frac{1}{2}n + \frac{1}{2},

(iii) μ_{\{0,i\}}^{4}(n) = \frac{1}{96}(4n^{3} + 54n^{2} + 92n + 45 + 3(−1)^{n}),

(iv) \begin{matrix}μ_{\{0,i\}}^{6}(n) & = \frac{1}{5120}(46n^{5} + 890n^{4} + 6320n^{3} + 11200n^{2} + 7789n + 2455 \\ & +15(−1)^{n}(23 + 29n) − 60i^{n}(2 + 3i − in) + 60(−i)^{n}(−2 + 3i + in)), \end{matrix}

where i^{2} = −1.

5. Further results

In this section we suggest several directions to generalize the results of the previous sections.

5.1. Average norms of T -polynomials with weights. The first of these directions is to obtain an exact formula for the average L_{2α}-norm over T -polynomials of degree exactly n with weight z^{m} for given α, n, and m. Let us define,

(8)  μ_{T}^{α}(n; m) = e_{T} (n; α/2, α/2, m) = \frac{1}{2πN_{T} (n)} ∑_{p∈𝔗_{n}} ∫_{0}^{2π} e^{imθ}|p(e^{iθ})|^{α} dθ,

Clearly, μ_{T}^{α}(n) = μ_{T}^{α}(n; 0) for all n, and α > 0. Now, if considering the following problem: find μ_{T}^{2α}(n; m) for any α > 0, n ≥ 0, and m ∈ ℤ? then by Theorem 1.1 the following result is true.

Theorem 5.1. Let T = {x_{1}, x_{2}, …, x_{d}} be any finite set of complex numbers. The generating function ∑_{n≥0} ∑_{m∈ℤ} μ_{T}^{2α}(n; m)x^{n}w^{m} is given by

∑_{n≥1} ∑_{j_{1},...,j_{n}=1}^{d} ∑_{\begin{matrix}k_{1} + ⋯ + k_{n} = α \\ ℓ_{1} + ⋯ + ℓ_{n} = α \end{matrix}} ∏_{a=1}^{n}(x_{j_{a}}^{t_{a}} \overline{x_{j_{a}}}^{r_{a}})\binom{α}{k_{1}, …, k_{n}}\binom{α}{ℓ_{1}, …, ℓ_{n}} \frac{x^{n−1}w^{( ∑_{a=1}^{n}(a−1)(r_{a}−t_{a}))}}{d^{n}} .

Moreover, μ_{T}^{2α}(n; m) is given by

\frac{1}{d^{n+1}} ∑_{j_{1},...,j_{n+1}=1}^{d} ∑_{\begin{matrix}k_{1} + ⋯ + k_{n+1} = α \\ ℓ_{1} + ⋯ + ℓ_{n+1} = α \\ ∑_{s=1}^{n+1}(s − 1)(ℓ_{s} − k_{s}) = m \end{matrix}} ∏_{a=1}^{n+1}(x_{j_{a}}^{k_{a}} \overline{x_{j_{a}}}^{ℓ_{a}})\binom{α}{k_{1}, …, k_{n+1}}\binom{α}{ℓ_{1}, …, ℓ_{n+1}}.

5.2. Exact formulas. The second of these directions is to obtain an exact formula for μ_{T}^{2α}(n; m) for given α and m. By definitions, it is clear that α!^{2}μ_{T}^{2α}(n; m) is the coefficient of x^{n}w^{m} of E_{T}^{α,α}(x, w) (see Section 3). Therefore, similarly as in Section 3 we get the following result.

Corollary 5.2. Let T = {x_{1}, x_{2}, …, x_{d}} be any finite set of complex numbers. Then

μ_{T}^{2α}(n; m) = \{ \begin{matrix}\frac{n+1−|m|}{d^{2}}A_{T}^{0,1}A_{T}^{1,0} & m ≠ 0, −n ≤ m ≤ m \\ \frac{n+1}{d}A_{T}^{1,1} & m = 0 \\ 0 & otherwise. \end{matrix}

This result can be extended to the case of μ_{T}^{4} (n). Since the answers become very cumbersome, we present here only the simplest case when A_{T}^{1,0} = ∑_{j=1}^{d} x_{j} = 0.

Corollary 5.3. Let T = {x_{1}, x_{2}, …, x_{d}} be any finite set of complex numbers such that A_{T}^{1,0} = ∑_{j=1}^{d} x_{j} = 0. Then

μ_{T}^{2α}(n; m) = \{ \begin{matrix}\frac{n+1−|m/2|}{d^{2}}A_{T}^{0,2}A_{T}^{2,0} & m ≠ 0, −2n ≤ m ≤ 2n, m even \\ \frac{n+1}{d}A_{T}^{1,1} + \frac{4}{d^{2}} \binom{n+1}{2}(A_{T}^{1,1})^{2} & m = 0 \\ 0 & otherwise. \end{matrix}

5.3. Average integrals. The third of these directions is to consider the general case to find an explicit formula for e_{T} (n, s, t, m). Theorem 1.1 and μ-algorithm give a complete answer for the generating function for these numbers, and explicit formula for e_{T} (n, s, t, m). For example, the following result is true. Corollary 5.4. Let T = {x_{1}, x_{2}, …, x_{d}} be any finite set of complex numbers. Then for any n ≥ 0 and m ∈ ℤ,

e_{T} (n, 1, 2, m) = \{ \begin{matrix}\frac{1}{d}A_{T}^{1,2} & 0 ≤ m ≤ n \\ 0 & otherwise \end{matrix}  and  e_{T} (n, 2, 1, m) = \{ \begin{matrix}\frac{1}{d}A_{T}^{2,1} & −n ≤ m ≤ 0 \\ 0 & otherwise \end{matrix} .

Acknowledgments. The author is grateful to the P. Borwein and K.S. Choi for the careful reading of the manuscript. The final version of the paper was written during the author visit at Haifa University, Israel. The author want to express his gratitude to Haifa University and A. Vainshtein for the support.

References

[B1] P. Borwein, Some old problems on polynomials with integer coefficients, to appear.

[B2] P. Borwein, Average norm of polynomials, Talk in 2001, avaiable in http://www/secm.sfu.ca/∼pborwein.

[B3] P. Borwein, Computational Excursions in analysis and number theory, CMS books in Mathematics, Springer-Verlag, New York, 2002.

[BC] P. Borwein and K.S. Choi, The average norm of polynomials of fixed height, to appear.

[L] J.E. Littlewood, Some problems in real and complex analysis, Heath Mathematical Monographs, Lexington, Mass, 1968.

[NB] D.J. Newman and J.S. Byrnes, The L^{4} norm of a polynomials with unimodular coefficients, Recent Advances in Fourier Analysis and its Applications, Kluwer (1990) 79–81.