On the generalized hypergeometric function, Sobolev orthogonal polynomials and biorthogonal rational functions

TL;DR

This paper explores the properties of partial sums of generalized hypergeometric series, revealing their connection to Sobolev orthogonal polynomials, biorthogonal rational functions, and related differential and recurrence relations.

Contribution

It establishes that partial sums of generalized hypergeometric series are Sobolev orthogonal polynomials and links them to biorthogonal rational functions, differential equations, and Jacobi-type pencils.

Findings

Partial sums are Sobolev orthogonal polynomials.

Polynomials relate to biorthogonal rational functions.

Connections to Jacobi-type pencils and differential equations.

Abstract

It turned out that the partial sums $g_n(z) = \sum_{k=0}^n \frac{(a_1)_k ... (a_p)_k}{(b_1)_k ... (b_q)_k} \frac{z^k}{k!}$, of the generalized hypergeometric series ${}_p F_q(a_1,...,a_p; b_1,...,b_q;z)$, with parameters $a_j,b_l\in\mathbb{C}\backslash\{ 0,-1,-2,... \}$, are Sobolev orthogonal polynomials. The corresponding monic polynomials $G_n(z)$ are polynomials of $R_I$ type, and therefore they are related to biorthogonal rational functions. Polynomials $g_n$ possess a differential equation (in $z$), and a recurrence relation (in $n$). We study integral representations for $g_n$, and some other their basic properties. Partial sums of arbitrary power series with non-zero coefficients are shown to be also related to biorthogonal rational functions. We obtain a relation of polynomials $g_n(z)$ to Jacobi-type pencils and their associated polynomials.