q-Type Lidstone expansions and an interpolation problem for entire functions

In this paper, we expand functions of specific q-exponential growth in terms its even (odd) Askey-Wilson q-derivatives at 0 and η=(q1/4+q−1/4)/2. This expansion is a q-version the celebrated Lidstone theorem, where function q-analogs polynomials, i.e., q-Bernoulli q-Euler polynomials as classical case. We also raise solve q-extension problem representing an entire form f(z)=g(z+1)−g(z), g(z) same order f(z).