Abstract. Recently, Liu et al. [Bilateral generating functions for the Chan-Chyan-Srivastava polynomials and the generalized Lauricella function, Integral Transform Spec. Funct. 23 (2012), no. 7, 539–549] investigated, in several interesting papers, some various families of bilateral generating functions involving the Chan-Chyan-Srivastava polynomials. The aim of this present paper is to obtain...

—An explicit formula for the Fourier coef cients of the Legendre polynomials can be found in the Bateman Manuscript Project. However, formulas for more general classes of orthogonal polynomials do not appear to have been worked out. Here we derive explicit formulas for the Fourier series of Gegenbauer, Jacobi, Laguerre and Hermite polynomials. The methods described here apply in principle to an...

We study a family of polynomials that are orthogonal with respect to the weight function e in [−1, 1], where ω ≥ 0. Since this weight function is complex-valued and, for large ω, highly oscillatory, many results in the classical theory of orthogonal polynomials do not apply. In particular, the polynomials need not exist for all values of the parameter ω, and, once they do, their roots lie in th...

We present a purely kinematic model of wave propagation in an excitable medium, namely cardiac tissue. The kinematic model is constructed from a standard reaction-diffusion model, using asymptotic techniques to track the position and velocity of each propagating wave front and wave back. The kinematic model offers a substantial improvement in computational efficiency without sacrificing the abi...

The second author dedicates this paper to the one and only Dick Askey, his mathematical father. Abstract. Three specializations of a set of orthogonal polynomials with " 8 different q's " are given. The polynomials are identified as q-analogues of Laguerre polynomi-als, and the combinatorial interpretation of the moments give infinitely many new Mahonian statistics on permutations.

We prove that if f is a real entire function of infinite order, then ff ′′ has infinitely many non-real zeros. In conjunction with the result of Sheil-Small for functions of finite order this implies that if f is a real entire function such that ff ′′ has only real zeros, then f is in the Laguerre-Pólya class, the closure of the set of real polynomials with real zeros. This result completes a l...

In some recent investigations involving differential operators for generalized Laguerre polynomials, Herman Bavinck (1996) encountered and proved a certain summation formula for the classical Laguerre polynomials. The main object of this sequel to Bavinck’s work is to prove a generalization of this summation formula for a class of hypergeometric polynomials. The demonstration, which is presente...

Random matrix model for the Nakagami-q (Hoyt) fading in multiple-input multiple-output (MIMO) communication channels with arbitrary number of transmitting and receiving antennas is considered. The joint probability density for the eigenvalues of HH (or HH), where H is the channel matrix, is shown to correspond to the Laguerre crossover ensemble of random matrices and is given in terms of a Pfaf...

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be “large.” For a fixed α ∈ Q−Z<0, Filaseta and Lam have shown that the nth degree Generalized Laguerre Polynomial L (α) n (x) = ∑n j=0 ( n+α n−j ) (−x)/j! is irreducible for all large enough n. We use our criterion to show that, under these conditions, the Galois group of L (α) n (x) is...

In our previous papers, the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials and the Casoratian identities for the Askey-Wilson polynomial and its reduced form polynomials were presented. These identities are naturally derived through quantum mechanical formulation of the classical orthogonal polynomials; ordinary quantum mechanics for the former and discrete quantum mechan...