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On spectral identities involving Gegenbauer polynomials

  • Richard Olu AwonusikaEmail author
Original Research Paper
  • 4 Downloads

Abstract

The Gegenbauer coefficients \(c_{j}^{\ell }(\nu )\) (\(1\le j\le \ell ;\, \nu >-1/2\)) associated with the normalised Gegenbauer polynomials \(\mathscr {C}_k^{\nu }\) describe the Maclaurin heat coefficients \(b^{n}_{2\ell }\) (\(n,\ell \ge 1\)) and the associated spectral polynomials \(\widetilde{\mathscr {R}}^{\nu }_{\ell }\) of the n-dimensional spheres \(\mathbb {S}^{n}\) (\(n\ge 1\)) and the real projective spaces \(\mathbf {P}^{n}(\mathbb {R})\) (\(n\ge 1\)). In this paper we introduce and construct a new class of spectral polynomials \(\mathscr {R}^{\nu }_{\ell }\) associated with the product \(\mathsf {C}_{k_1,k_2}^{\nu }:=\mathscr {C}_{k_1}^{\nu }\times \mathscr {C}_{k_2}^{\nu }\) (\(k_{1},k_{2}\ge 0\); \(\nu >-1/2\)) and evaluate explicitly some definite integrals involving the Gengebauer polynomials \(C_{k}^{\nu }\) (\(k\ge 0, \nu >-1/2\)) in terms of these spectral polynomials. These integrals apart from being interesting in their own right lead to identities that are novel in the context of special functions.

Keywords

Gegenbauer coefficients Maclaurin heat coefficients Gegenbauer polynomials Special functions 

Mathematics Subject Classification

33C05 33C45 35A08 35C05 35C10 35C15 

Notes

Compliance with ethical standards

Conflict of interest

There is no conflict of interest.

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Copyright information

© Forum D'Analystes, Chennai 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAdekunle Ajasin UniversityAkungba AkokoNigeria

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