We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants γk(a). These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials Ln to investigate a certain sum Sγ(n) involv...

This paper presents a numerical method for the approximate solution of mthorder linear delay difference equations with variable coefficients under the mixed conditions in terms of Laguerre polynomials. The aim of this article is to present an efficient numerical procedure for solving mth-order linear delay difference equations with variable coefficients. Our method depends mainly on a Laguerre ...

Orthogonal polynomials have been used to produce sharp estimates in Harmonic Analysis in several instances. The first most notorious and original use was in Beckner’s thesis [1], where he proved the sharp Hausdorff-Young inequality using Hermite polynomial expansions. More recently, Foschi [4] used spherical harmonics and Gegenbauer polynomials in his proof of the sharp Tomas-Stein adjoint Four...

belong to one of the three family of orthogonal polynomials, the other two being Jacobi and Legendre. In addition to their important roles in mathematical analysis, these polynomials also feature prominently in algebra and number theory. Schur ([7], [8]) pioneered the study of Galois properties of specializations of these orthogonal polynomials, and Feit [1] used them to solve the inverse Galoi...

We formulate and duscuss two open problems. The rst one is due to PP olya. It states that the sequence of polynomials formed by a polynomial p with only real zeros and all its derivatives, obeys Descartes' rule of signs for any x, greater than the largest zero of p. The other problem is due to Karlin and states that certain Hankel determinants associated with an entire function in the Laguerre-...

The monomiality principle was introduced (see e.g. [1] and the references therein) in order to derive properties of special polynomials from the corresponding ones holding for monomials. A set of paramount importance in this framework is given by the Hermite-Kampé de Fériet (shortly H-KdF) or Gould-Hopper polynomials [2], [3], [4]. As it is well known, the Hermite-Kampé de Fériet polynomials H ...

The polynomial sets {Y"(x; k)} and { Z"(x; &)}, discussed by Joseph D. E. Konhauser, are biorthogonal over the interval (0, oo) with respect to the weight function x a e~ x , where a > — 1 and A: is a positive integer. The object of the present note is to develop a fairly elementary method of proving a general multilinear generating function which, upon suitable specializations, yields a number...

Three classes of higher-order hierarchical basis functions constructed from standard orthogonal polynomials are proposed and evaluated for modeling of combined metallic and dielectric microwave structures. The reduction of the condition number of moment-method matrices is several orders of magnitude when compared to the technique using regular polynomial basis functions.

Recent theoretical studies of chaotic scattering have encounted ensembles of random matrices in which the eigenvalue probability density function contains a one-body factor with an exponent proportional to the number of eigenvalues. Two such ensembles have been encounted: an ensemble of unitary matrices specified by the so-called Poisson kernel, and the Laguerre ensemble of positive definite ma...

The generalisation of continuous orthogonal polynomial ensembles from random matrix theory to the q-lattice setting is considered. We take up task initiating a systematic study corresponding moments density two complementary viewpoints. first requires knowledge ensemble average with respect general Schur polynomial, which spectral follow as corollary. In case little q-Laguerre weight, particula...