The q-Heun operator of big q-Jacobi type and the q-Heun algebra

TL;DR

This paper introduces the q-Heun operator of the big q-Jacobi type, explores its algebraic structure, and connects it to biorthogonal polynomials and eigenvalue problems within the q-Hahn algebra framework.

Contribution

It defines the most general second order q-difference operator of this type and extends the q-Hahn algebra to include it as a generator.

Findings

The q-Heun operator maps polynomials of degree n to n+1.

Biorthogonal Pastro polynomials satisfy a generalized eigenvalue problem involving q-Heun operators.

The paper also analyzes the special case related to little q-Jacobi polynomials.

Abstract

The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second order q-difference operator that maps polynomials of degree $n$ to polynomials of degree $n+1$. It is tridiagonal in bases made out of either q-Pochhammer or big q-Jacobi polynomials and is bilinear in the operators of the q-Hahn algebra. The extension of this algebra that includes the q-Heun operator as generator is described. Biorthogonal Pastro polynomials are shown to satisfy a generalized eigenvalue problem or equivalently to be in the kernel of a special linear pencil made out of two q-Heun operators. The special case of the q-Heun operator associated to the little q-Jacobi polynomials is also treated.