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

Sergey M. Zagorodnyuk

paper

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

arXiv:2101.04479

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 $a_{j}, b_{l} \in C \ {0, - 1, - 2, ...}$ , 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.

11 pages

References in corpus (1)

On modified kernel polynomials and classical type Sobolev orthogonal polynomials