We show that the discriminant of the generalized Laguerre polynomial L n (x) is a non-zero square for some integer pair (n, α), with n ≥ 1, if and only if (n, α) belongs to one of 30 explicitly given infinite sets of pairs or to an additional finite set of pairs. As a consequence, we obtain new information on when the Galois group of L n (x) over Q is the alternating group An. For example, we e...

A calculation of the linearization coefficients of the (generalized) Laguerre polynomials L")(x) is proposed by means of analytic and combinatorial methods. This paper extends to the case of an arbitrary a a combinatoric and analytic result due to Askey, Ismail, and Koornwinder and Even and Gillis. Theorem (/-extension) AMS(MOS) subject classifications. 33A65, 05A 15 1. Introduction. Let (p,,(x...

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.

Exceptional orthogonal polynomial systems (X-OPS) arise as eigenfunctions of Sturm-Liouville problems and generalize in this sense the classical families of Hermite, Laguerre and Jacobi. They also generalize the family of CPRS orthogonal polynomials introduced by Cariñena et al., [3]. We formulate the following conjecture: every exceptional orthogonal polynomial system is related to a classical...

For a class of orthogonal polynomials related to the q-Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted L-space explicitly. The result is proved in two ways; by a spectral decomposition of a suitable opera...

We show that the Meixner, Pollaczek, Meixner-Pollaczek and Al-Salam-Chihara polynomials, in certain normalization, are moments of probability measures. We use this fact to derive bilinear and multilinear generating functions for some of these polynomials. We also comment on the corresponding formulas for the Charlier, Hermite and Laguerre polynomials. Running Title: Generating Functions

We show that a certain generalized beta function B(x; y; b) which reduces to Euler’s beta functions B(x; y) when its variable b vanishes and preserves symmetry in its parameters may be represented in terms of a nite number of well known higher transcendental functions except (possibly) in the case when one of its parameters is an integer and the other is not. In the latter case B(x; y; b) may b...

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.

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...