Realizations of coupled vectors in the tensor product of representations of and

Using the realization of positive discrete series representations of in terms of a complex variable z, we give an explicit expression for coupled basis vectors in the tensor product of ν+1 representations as polynomials in ν+1 variables z₁,…,z_ν+1. These expressions use the terminology of binary coupling trees (describing the coupled basis vectors), and are explicit in the sense that there is no reference to the Clebsch–Gordan coefficients of . In general, these polynomials can be written as (terminating) multiple hypergeometric series. For ν=2, these polynomials are triple hypergeometric series, and a relation between the two binary coupling trees yields a relation between two triple hypergeometric series. The case of is discussed next. Also here the polynomials are determined explicitly in terms of a known realization; they yield an efficient way of computing coupled basis vectors in terms of uncoupled basis vectors.     

Linearization of the product of symmetric orthogonal polynomials

A new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials. We use the monic three-term recurrence relation of an orthogonal polynomial system to set up a partial difference equation problem for the product of two polynomials and solve it in terms of the initial data. To this end, an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function. As an application, we derive the linearization formulas for the associated Hermite polynomials and for their continuousq-analogues. The linearization coefficients are represented here in terms of₃ F ₂ and₃Φ₂ (basic) hypergeometric functions, respectively. We also give some partial results in the case of the associated continuousq-ultraspherical polynomials.     

On the zeros ofζ (1)(s) — a (on the zeros of a class of a generalized Dirichlet series — XVII)

Some very precise results (see Theorems 4 and 5) are proved about thea-values of thelth derivative of a class of generalized Dirichlet series, forl≥l _o =l _o(a) (l _o being a large constant). In particular for the precise results on the zeros ofζ ⁽¹⁾ (s) —a (a any complex constant andl≥l _o) see Theorems 1 and 2 of the introduction.     

Computation of the canonical polynomials and applications to some optimal control problems

The Tau method is a numerical technique that consists in constructing polynomial approximate solutions for ordinary differential equations. This method has two approaches: operational and recursive. The former converts the differential problem to a matrix problem and produces approximations in terms of a prescribed orthogonal polynomials basis. In the recursive approach, we construct approximate solutions in terms of a special set of polynomials {Q _k (t); k?=?0, 1, 2...} called canonical polynomials basis. In some cases, the Q _k ??s can be obtained explicitly through a recursive formula. But no analogous formulae are reported in the literature for the general cases. In this paper, utilizing the operational Tau method, we develop an algorithm that allows to generate those canonical polynomials iteratively and explicitly. In addition, we demonstrate the capability of the operational Tau method in treating quadratic optimal control problems governed by ordinary differential equations.     

Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations

We find an explicit combinatorial interpretation of the coefficients of Kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups S(n) in terms of free cumulants R₂,R₃,… of the corresponding Young diagram. Our interpretation is based on counting certain factorizations of a given permutation.     

The Betti polynomials of powers of an ideal

For an ideal I in a regular local ring or a graded ideal I in the polynomial ring we study the limiting behavior of as k goes to infinity. By Kodiyalam’s result it is known that β_i(S/I^k) is a polynomial for large k. We call these polynomials the Kodiyalam polynomials and encode the limiting behavior in their generating polynomial. It is shown that the limiting behavior depends only on the coefficients on the Kodiyalam polynomials in the highest possible degree. For these we exhibit lower bounds in special cases and conjecture that the bounds are valid in general. We also show that the Kodiyalam polynomials have weakly descending degrees and identify a situation where the polynomials all have the highest possible degree.     

On the exact Markov inequality for k-monotone polynomials in uniform and L 1-norms

We consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the Markov inequality for monotone polynomials. In this note we give Markov-type inequalities for higher order derivatives in the general class of k-monotone polynomials. In particular, in case of first derivative we find the exact solution of this extremal problem in both uniform and L ₁-norms. This exact solution is given in terms of the largest zeros of certain Jacobi polynomials.     

On the Littlewood cyclotomic polynomials

In this article, we study the cyclotomic polynomials of degree N−1 with coefficients restricted to the set {+1,−1}. By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p(x) with coefficients ±1 of even degree N−1 is cyclotomic if and only if p(x)=±Φp₁(±x)Φp₂(±xp₁)?Φpr(±xp₁p₂?pr−1), where N=p₁p₂?pr and the pi are primes, not necessarily distinct. Here is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree α²pβ−1 with odd prime p or separable polynomials of any odd degree.     

An identity involving partitional generalized binomial coefficients

Define coefficients (_κ^λ) by C_λ(I_p + Z)/C_λ(I_p) = Σ_k=0^l Σ_{?∈$\text{P}$k} (_?^λ) C_κ(Z)/C_κ(I_p), where the C_λ's are zonal polynomials in p by p matrices. It is shown that C_?(Z) etr(Z)/k! = Σ_l=k^∞ Σ_{λ∈$\text{P}$l} (_?^λ) C_λ(Z)/l!. This identity is extended to analogous identities involving generalized Laguerre, Hermite, and other polynomials. Explicit expressions are given for all (_?^λ), ? ∈ $\text{P}$_k, k ≤ 3. Several identities involving the (_?^λ)'s are derived. These are used to derive explicit expressions for coefficients of $\text{C}{}_{\mathrm{\lambda }}\text{(Z)}\text{l}\text{!}$ in expansions of P(Z), $\text{etr}\text{(Z)}\text{k!}$ for all monomials P(Z) in s_j = tr Z^j of degree k ≤ 5.     

Approximation of integrable functions by general linear operators of their fourier series

The pointwise estimates of the deviation T n,A,B f (·) f (·) in terms of moduli of continuity .f and w.f there are proved. Analogical results on norm approximation with remarks and corollaries are also given. In the results there are used the essentially weaker conditions than these in [Mittal, M. L.: J. Math. Anal. Appl., 220, 434-450 (1998) Theorem 1, p. 437].     

The spectrum of Jacobi matrices in terms of its associated weight function

Using the so-called Lanczos procedure of orthogonalization a method is developed to calculate the elements of a N-dimensional Jacobi matrix and/or the coefficients of the three-term recurrence relation of a system of orthogonal polynomials {P_m(x), m = 0, 1, 2, ?, N} in terms of the moments μ_r^′(1) of its associated weight function. The eigenvalue density ?^(N)(x) and its asymptotical limit, i.e. when N tends to infinite, are also calculated in terms of μ_r^′(1). The method is used to determine the functions ?^(N)(x) and ?(x) for some known weight functions, like the normal distribution, the uniform distribution, the semicircular distribution and the gamma or Pearson type III distribution. As a byproduct the asymptotical density of zeros of Chebyshev, Legendre and generalized Laguerre polynomials are found.     

Representations and the Jacobian Conjecture

In this paper we apply the representation theory of the Lie algebra sl₂(C) to the problem of describing Hessian nilpotent polynomials, which are important in the theory of the Jacobian Conjecture. In the two variable case we describe them as the maximal and minimal weight vectors of the irreducible representations of sl₂(C). For the first time this gives a characterization of the Hessian nilpotent polynomials in terms of linear differential operators.     

Some Inequalities for Derivatives of Trigonometric and Algebraic Polynomials

We investigate inequalities for derivatives of trigonometric and algebraic polynomials in weighted L ^P spaces with weights satisfying the Muckenhoupt A _p condition. The proofs are based on an identity of Balázs and Kilgore [1] for derivatives of trigonometric polynomials. Also an inequality of Brudnyi in terms of rth order moduli of continuity ω_r will be given. We are able to give values to the constants in the inequalities.     

Chain sequences and symmetric generalized orthogonal polynomials

In this paper, we derive an explicit expression for the parameter sequences of a chain sequence in terms of the corresponding orthogonal polynomials and their associated polynomials. We use this to study the orthogonal polynomials K_n^(λ,M,k) associated with the probability measure dφ(λ,M,k;x), which is the Gegenbauer measure of parameter λ+1 with two additional mass points at ±k. When k=1 we obtain information on the polynomials K_n^(λ,M) which are the symmetric Koornwinder polynomials. Monotonicity properties of the zeros of K_n^(λ,M,k) in relation to M and k are also given.     

Construction of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines

The paper presents a simple procedure for the construction of quasi-interpolation operators in spaces of m-harmonic splines in R^d, which reproduce polynomials of high degree. The procedure starts from a generator ?₀, which is easy to derive but with corresponding quasi-interpolation operator reproducing only linear polynomials, and recursively defines generators ?₁,?₂,…,?_m−1 with corresponding quasi-interpolation operators reproducing polynomials of degree up to 3,5,…,2m−1 respectively. The construction of ?_j from ?_j−1 is explicit, simple and independent of m. The special case d=1 and the special cases d=2,m=2,3,4 are discussed in details.     

Hypergeometric series solutions of linear operator equations

Let K be a field and L:K[x]→K[x] be a linear operator acting on the ring of polynomials in x over the field K. We provide a method to find a suitable basis {b_k(x)} of K[x] and a hypergeometric term c_k such that is a formal series solution to the equation L(y(x))=0. This method is applied to construct hypergeometric representations of orthogonal polynomials from the differential/difference equations or recurrence relations they satisfied. Both the ordinary cases and the q-cases are considered.     

A result on common zeros location of orthogonal polynomials in two variables

The aim of this paper is to investigate some general properties of common zeros of orthogonal polynomials in two variables for any given region D⊂R² from a view point of invariant factor. An important result is shown that if X₀ is a common zero of all the orthogonal polynomials of degree k then the intersection of any line passing through X₀ and D is not empty. This result can be used to settle the problem of location of common zeros of orthogonal polynomials in two variables. The main result of the paper can be considered as an extension of the univariate case.     

Coordinatewise multiple summing operators in Banach spaces

We invent the new notion of coordinatewise multiple summing operators in Banach spaces, and use it to study various vector valued extensions of the well-know Bohnenblust-Hille inequality (which originally extended Littlewood's 4/3-inequality). Our results have application on the summability of monomial coefficients of m-homogeneous polynomials P:?_∞→?p, as well as for the convergence theory of products of vector valued Dirichlet series.     

On series containing products of Legendre polynomials

Explicit formulas are established for infinite sums of products of three or four Legendre polynomials of nth order with coefficients 2n + 1; the series depends only the arguments of the polynomials and contains no other variables. We show that, for the product of three polynomials, the sum is inverse to the root of the product of four sine functions and, in the case of four polynomials, this expression additionally contains the elliptic integral K(k) as a multiplier. Analogs and particular cases are considered which allow one to compare the relationships proved in this note with results proved in various domains of mathematical physics and classical functional analysis.     

Multiple Orthogonal Polynomials Associated with the Modified Bessel Functions of the First Kind

Abstract. We consider polynomials which are orthogonal with respect to weight functions, which are defined in terms of the modified Bessel function I _ν and which are related to the noncentral χ ² -distribution. It turns out that it is the most convenient to use two weight functions with indices ν and ν+1 and to study orthogonality with respect to these two weights simultaneously. We show that the corresponding multiple orthogonal polynomials of type I and type II exist and give several properties of these polynomials (differential properties, Rodrigues formula, explicit formulas, recurrence relation, differential equation, and generating functions).