Mixed-type hypergeometric Bernoulli-Gegenbauer polynomials: some properties
type: Item
instance of: scholarly article
title: Mixed-type hypergeometric Bernoulli-Gegenbauer polynomials: some properties
author: Dionisio Peralta
author: Yamilet Quintana
published in: Communications in Applied and Industrial Mathematics
publication date: 2024-12-10T00:00:00Z
review text: The purpose of this article is to study mixed-type hypergeometric Bernoulli-Gegenbauer polynomials by using the generating function. Some well-known notations for number systems are given. It is not necessary to state that \(0^0=1\), since this is a standard convention. The corresponding differential equation for hypergeometric Bernoulli-Gegenbauer polynomials is referred to. The standard generating function for Gegenbauer polynomials which occurs after formula (10) can be found in \textit{E. D. Rainville}'s book [Special functions. Bronx, N. Y.: Chelsea Publishing Company (1971; Zbl 0231.33001), p. 277], and not in the cited [\textit{G. Szegö}, Orthogonal polynomials. 4th ed. Providence, RI: American Mathematical Society (AMS) (1975; Zbl 0305.42011)]. On the other hand, the generating functions for the monic Gegenbauer polynomials (9) and (10) are not proved. The orthogonality relation for Gegenbauer polynomials with a new weight function on page 5 cannot be found in [{G. Szegö}, loc. cit.; Chapter IV]. Then diagrams for the zeros of the new polynomials are given. I have tried to correct some of the statements in the paper. To sum up, it would have been better to follow {E. D. Rainville}'s book [loc. cit.] and his notation for Gegenbauer polynomials.
reviewed by: Thomas Ernst
Mathematics Subject Classification ID: 33E20
Mathematics Subject Classification ID: 33E30
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 11B68
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 7956572
zbMATH Keywords: Gegenbauer polynomials
zbMATH Keywords: generalized Bernoulli polynomials
zbMATH Keywords: hypergeometric Bernoulli polynomials
zbMATH Keywords: operational methods
MaRDI profile type: Publication
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DOI: 10.2478/CAIM-2024-0018
description: scientific article; zbMATH DE number 7956572