On some hypergeometric Sobolev orthogonal polynomials with several continuous parameters

Abstract

In this paper we study the following hypergeometric polynomials: $\mathcal{P}_n(x) = \mathcal{P}_n(x;\alpha,\beta,\delta_1,\dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = {}_{\rho+2} F_{\rho+1} (-n,n+\alpha+\beta+1,\delta_1+1,\dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x)$, and $\mathcal{L}_n(x) = \mathcal{L}_n(x;\alpha,\delta_1,\dots,\delta_\rho,\kappa_1,\dots,\kappa_\rho) = {}_{\rho+1} F_{\rho+1} (-n,\delta_1+1,\dots,\delta_\rho+1;\alpha+1,\kappa_1+\delta_1+1,\dots,\kappa_\rho+\delta_\rho+1;x)$, $n\in\mathbb{Z}_+$, where $\alpha,\beta,\delta_1,\dots,\delta_\rho\in(-1,+\infty)$, and $\kappa_1,\dots,\kappa_\rho\in\mathbb{Z}_+$, are some parameters. The natural number $\rho$ of the continuous parameters $\delta_1,\dots,\delta_\rho$ can be chosen arbitrarily large. It is seen that the special case $\kappa_1=\dots=\kappa_\rho=0$ leads to Jacobi and Laguerre orthogonal polynomials. In general, it is shown that polynomials $\mathcal{P}_n(x)$ and $\mathcal{L}_n(x)$ are Sobolev orthogonal polynomials on the real line with some explicit matrix measures. We study integral representations, differential equations and generating functions for these polynomials. Recurrence relations and properties of their zeros are discussed as well.