On spectral identities involving Gegenbauer polynomials


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.

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Correspondence to Richard Olu Awonusika.

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Awonusika, R.O. On spectral identities involving Gegenbauer polynomials. J Anal 27, 1123–1137 (2019). https://doi.org/10.1007/s41478-019-00163-7

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  • Gegenbauer coefficients
  • Maclaurin heat coefficients
  • Gegenbauer polynomials
  • Special functions

Mathematics Subject Classification

  • 33C05
  • 33C45
  • 35A08
  • 35C05
  • 35C10
  • 35C15