A set of orthogonal polynomials with 8 independent “q’s” is defined which generalizes the Laguerre polynomials. The moments of the measure for these polynomials are the generating functions for permutations according to eight different statistics. Specializing these statistics gives many other well-known sets of combinatorial objects and relevant statistics. The specializations are studied, wit...

In this note, we have obtained a novel extension of a bilateral generating relationsinvolvingbiorthogonal polynomials ; from the existence of quasi-bilinear generating relation by group theoretic method. As particular cases, we obtain the corresponding results on generalised Laguerre polynomials.

In this paper we address the problem of approximating functions on a semi-in nite interval by truncated series of orthonormal generalized Laguerre functions. The generalized Laguerre functions contain two parameters, namely a scale factor and an order of generalization. The rate of convergence of a generalized Laguerre series depends on the choice of these parameters. Results concerning the det...

New inequalities are investigated for real entire functions in the Laguerre-Pólya class. These are generalizations of the classical Turán and Laguerre inequalities. They provide necessary conditions for certain real entire functions to have only real zeros.

A simple structure of the multiple Laguerre polynomial expansions is used to study the Cess aro summability above the critical index for the convolution type Laguerre expansions. The multiple Laguerre polynomial expansion of an`1-radial function f 0 (jxj) is shown to be an`1-radial function that coincides with the Laguerre polynomial expansion of f 0 , which allows us to settle the problem of s...

The rational solutions for the discrete Painlevé II equation are constructed based on the bilinear formalism. It is shown that they are expressed by the determinant whose entries are given by the Laguerre polynomials. Continuous limit to the Devisme polynomial representation of the rational solutions for the Painlevé II equation is also discussed.

We exploit the concepts and the formalism associated with the principle of monomiality to derive the ortogonality properties of the associated Laguerre polynomials. We extend a recently developed technique of algebraic nature and comment on the usefulness of the proposed method.

Let φ and f be functions in the Laguerre–Pólya class. Write φ(z) = e−αzφ1(z) and f (z) = e−βzf1(z), where φ1 and f1 have genus 0 or 1 and α,β 0. If αβ < 1/4 and φ has infinitely many zeros, then φ(D)f (z) has only simple real zeros, where D denotes differentiation.  2004 Elsevier Inc. All rights reserved.

We establish a large n complete asymptotic expansion for q-Laguerre polynomials and a complete asymptotic expansion for a q-Bessel function of large argument. These expansions are needed in our study of an exactly solvable random transfer matrix model for disordered electronic systems. We also give a new derivation of an asymptotic formula due to Littlewood (1907).

0. Introduction 1. Borel decomposition and the 2-Toda lattice 2. Two-Toda τ -functions and Pfaffian τ̃ -functions 3. The Pfaffian Toda lattice and skew-orthogonal polynomials 4. The (s = −t)-reduction of the Virasoro vector fields 5. A representation of the Pfaffian τ̃ -function as a symmetric matrix integral 6. String equations and Virasoro constraints 7. Virasoro constraints with boundary terms...