Two Finite q-Sturm-Liouville Problems and their Orthogonal Polynomial Solutions

Abstract

In this paper, we consider two new $q$-Sturm-Liouville problems and prove that their polynomial solutions are finitely orthogonal with respect to two weight functions which correspond to Fisher and T-student distributions as $q\to 1$. Then, we obtain the general properties of these polynomial solutions, such as orthogonality relations, three term recurrence relations, $q$-difference equations and basic hypergeometric representations, where all results in the continuous case are recovered as $q\to 1$.