$q$-type Lidstone expansions and an interpolation 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 $f(z)$.