Connection relations and bilinear formulas for the classical orthogonal polynomials

AbstractSuppose we have an operator T that maps a set of orthogonal polynomials {Pn(x)}n = o∞ to another set of orthogonal polynomials. We show how such a mapping can be used to derive connection relations as well as bilinear formulas for the pre-images {Pn(x)}n = o∞. This method is carried out in detail for the Jacobi, Laguerre, and Hahn polynomials

Differential coefficients of orthogonal matrix polynomials

AbstractWe find explicit formulas for raising and lowering first order differential operators for orthogonal matrix polynomials. We derive recurrence relations for the coefficients in the raising and lowering operators. Some examples are given

Variants of the Rogers–Ramanujan Identities

AbstractWe evaluate several integrals involving generating functions of continuous q-Hermite polynomials in two different ways. The resulting identities give new proofs and generalizations of the Rogers–Ramanujan identities. Two quintic transformations are given, one of which immediately proves the Rogers–Ramanujan identities without the Jacobi triple product identity. Similar techniques lead to new transformations for unilateral and bilateral series. The quintic transformations lead to curious identities involving primitive fifth roots of unity which are then extended to primitive pth roots of unity for odd p

Asymptotics of basic Bessel functions and q-Laguerre polynomials

AbstractWe 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)

Some Orthogonal Polynomials Related to Elliptic Functions

AbstractWe characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates

Tau-Function Constructions of the Recurrence Coefficients of Orthogonal Polynomials

AbstractIn this paper we compute the recurrence coefficients of orthogonal polynomials using τ-function techniques. It is shown that for polynomials orthogonal with respect to positive weight functions on a noncompact interval, the recurrence coefficient can be expressed as the change in the chemical potential which, for sufficiently largeNis the second derivative of the free energy with respect toN, the particle number. We give three examples using this technique: Freud weights, Erdős weights, and weak exponential weights

On Formulas Of Ramanujan And Evans

We give short elementary proofs of a formula of Ramanujan as interpreted by Bradley, and a companion formula originally proved by W. Chu. We also give an elementary proof of a generalization of an identity originally proved using modular functions and used to study a generating function for the number of partitions with specified crank. © Springer Science + Business Media, LLC 2006