$q$-type Lidstone expansions and an interpolation problem for entire functions

TL;DR

This paper develops a $q$-analogue of the Lidstone expansion theorem for functions with specific $q$-exponential growth, using $q$-derivatives and polynomials, and addresses a $q$-extension of a classical entire function representation problem.

Contribution

It introduces a $q$-version of the Lidstone expansion theorem and solves a $q$-analogue of an entire function representation problem.

Findings

Established a $q$-Lidstone expansion for functions with $q$-exponential growth.

Derived a $q$-interpolation formula for entire functions of a specific form.

Solved the $q$-extension of the representation problem for entire functions.

Abstract

In this paper, we expand functions of specific $q$-exponential growth in terms of its even (odd) Askey- Wilson $q$-derivatives at $0$ and $\eta=(q^{1/4}+q^{-1/4})/2$. This expansion is a $q$-version of the celebrated Lidstone expansion theorem, where we expand the function in $q$-analogs of Lidstone polynomials, i.e., q-Bernoulli and $q$-Euler polynomials as in the classical case. We also raise and solve a $q$-extension of the problem of representing an entire function of the form $f(z)=g(z+1)-g(z)$, where $g(z)$ is also an entire function of the same order as $f(z)$.