On spectral identities involving Gegenbauer polynomials

  • Richard Olu AwonusikaEmail author
Original Research Paper


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.


Gegenbauer coefficients Maclaurin heat coefficients Gegenbauer polynomials Special functions 

Mathematics Subject Classification

33C05 33C45 35A08 35C05 35C10 35C15 


Compliance with ethical standards

Conflict of interest

There is no conflict of interest.


  1. 1.
    Awonusika, R.O., and A. Taheri. 2017. On jacobi polynomials \((\mathscr {P}_k^{(\alpha, \beta )}: \alpha, \beta >-1)\) and Maclaurin spectral functions on rank one symmetric spaces. The Journal of Analysis 25: 139–166.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Awonusika, R.O., and A. Taheri. 2017. On Gegenbauer polynomials and coefficients \(c^{\ell }_{j}(\nu )\) (\(1\le j\le \ell\), \(\nu >-1/2\)). Results in Mathematics 72: 1359–1367.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Awonusika, R.O. 2018. Special function representations of the Poisson kernel on hyperbolic spaces. Journal of Mathematical Chemistry 56: 825–849.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Awonusika, R.O., and A. Taheri. 2018. A spectral identity on Jacobi polynomials and its analytic implications. Canadian Mathematical Bulletin 61: 473–482.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gradshtejn, I.S., and I.M. Ryzhik. 2007. Table of integrals, series and products. Cambridge: Academic Press.Google Scholar
  6. 6.
    Mueller, C.E., and F.B. Weissler. 1982. Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the \(n\)-sphere. Journal of Functional Analysis 48: 252–283.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Vilenkin, N.J. 1968. Special functions and the theory of group representations. Translations of mathematical monographs, vol. 22, AMS.Google Scholar
  8. 8.
    Dijksma, A., and T. Koornwinder. 1971. Spherical harmonics and the product of two Jacobi polynomials. Indagationes Mathematicae 33: 191–196.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Koornwinder, T. 1974. Jacobi polynomials, II. An analytic proof of the product formula. SIAM Journal on Mathematical Analysis 5: 125–137.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Forum D'Analystes, Chennai 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAdekunle Ajasin UniversityAkungba AkokoNigeria

Personalised recommendations