An algebraic treatment of the Pastro polynomials on the real line

TL;DR

This paper explores the algebraic structure and properties of Pastro polynomials on the real line, revealing their q-difference equations, recurrence relations, and biorthogonality through operator methods.

Contribution

It introduces an algebraic framework using q-difference operators to analyze Pastro polynomials, deriving their fundamental equations and orthogonality relations.

Findings

Pastro polynomials satisfy specific q-difference equations and recurrence relations.

A discrete biorthogonality relation for Pastro polynomials is established.

The algebra generated by the q-difference operators is characterized.

Abstract

The properties of the Pastro polynomials on the real line are studied with the help of a triplet of $q$-difference operators. The $q$-difference equation and recurrence relation these polynomials obey are shown to arise as generalized eigenvalue problems involving the triplet of operators, with the Pastro polynomials as solutions. Moreover, a discrete biorthogonality relation on the real line for the Pastro polynomials is obtained and is then understood using adjoint operators. The algebra realized by the triplet of $q$-difference operators is investigated.