On spectral identities involving Gegenbauer polynomials

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.

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References

  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.

    Google 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.

    Google Scholar 

  3. 3.

    Awonusika, R.O. 2018. Special function representations of the Poisson kernel on hyperbolic spaces. Journal of Mathematical Chemistry 56: 825–849.

    MathSciNet  Article  Google 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.

    MathSciNet  Article  Google 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.

    Google Scholar 

  7. 7.

    Vilenkin, N.J. 1968. Special functions and the theory of group representations. Translations of mathematical monographs, vol. 22, AMS.

  8. 8.

    Dijksma, A., and T. Koornwinder. 1971. Spherical harmonics and the product of two Jacobi polynomials. Indagationes Mathematicae 33: 191–196.

    MathSciNet  Article  Google Scholar 

  9. 9.

    Koornwinder, T. 1974. Jacobi polynomials, II. An analytic proof of the product formula. SIAM Journal on Mathematical Analysis 5: 125–137.

    MathSciNet  Article  Google Scholar 

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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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Keywords

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

Mathematics Subject Classification

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