# On some classical type Sobolev orthogonal polynomials

## TL;DR

This paper introduces a method to construct classical type Sobolev orthogonal polynomials using hypergeometric functions, generalizing Laguerre and Jacobi polynomials, and explores their properties and differential equations.

## Contribution

The paper presents explicit constructions of Sobolev orthogonal polynomials from hypergeometric functions and derives their fundamental properties and differential equations.

## Key findings

- Polynomials satisfy higher-order differential equations.
- Explicit measures for orthogonality are provided for positive integer parameters.
- Recurrence relations and basic properties are established.

## Abstract

In this paper we propose a way to construct classical type Sobolev orthogonal polynomials. We consider two families of hypergeometric polynomials: ${}_2 F_2(-n,1;q,r;x)$ and ${}_3 F_2(-n,n-1+a+b,1;a,c;x)$ ($a,b,c,q,r>0$, $n=0,1,...$), which generalize Laguerre and Jacobi polynomials, respectively. These polynomials satisfy higher-order differential equations of the following form: $L y + \lambda_n D y = 0$, where $L,D$ are linear differential operators with polynomial coefficients not depending on $n$. For positive integer values of the parameters $r,c$ these polynomials are Sobolev orthogonal polynomials with some explicitly given measures. Some basic properties of these polynomials, including recurrence relations, are obtained.