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 η=(q1/4+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).