On some hypergeometric Sobolev orthogonal polynomials with several continuous parameters

TL;DR

This paper introduces a family of hypergeometric Sobolev orthogonal polynomials with multiple parameters, generalizing classical Jacobi and Laguerre polynomials, and explores their properties, representations, and orthogonality measures.

Contribution

It defines new hypergeometric Sobolev orthogonal polynomials with several continuous parameters and analyzes their integral representations, differential equations, generating functions, and zero properties.

Findings

Polynomials are Sobolev orthogonal with explicit matrix measures.

Derived integral representations, differential equations, and generating functions.

Analyzed recurrence relations and zero distributions.

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.