Algebraic generating functions for Gegenbauer polynomials

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

It is shown that several of Brafman's generating functions for the Gegenbauer polynomials are algebraic functions of their arguments if the Gegenbauer parameter differs from an integer by one-fourth or one-sixth. Two examples are given, which come from recently derived expressions for associated Legendre functions with tetrahedral or octahedral monodromy. It is also shown that if the Gegenbauer parameter is restricted as stated, the Poisson kernel for the Gegenbauer polynomials can be expressed in terms of complete elliptic integrals. An example is given.

Original languageEnglish (US)
Title of host publicationFrontiers In Orthogonal Polynomials and Q-series
PublisherWorld Scientific Publishing Co. Pte Ltd
Pages425-444
Number of pages20
Volume1
ISBN (Electronic)9789813228887
ISBN (Print)9789813228870
DOIs
StatePublished - Jan 12 2018

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Gegenbauer Polynomials
Algebraic function
Generating Function
Poisson Kernel
Legendre function
Elliptic integral
Monodromy
Integer

Keywords

  • Gegenbauer polynomial
  • Generating function
  • Hypergeometric transformation
  • Legendre function
  • Legendre polynomial
  • Poisson kernel

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Maier, R. S. (2018). Algebraic generating functions for Gegenbauer polynomials. In Frontiers In Orthogonal Polynomials and Q-series (Vol. 1, pp. 425-444). World Scientific Publishing Co. Pte Ltd. https://doi.org/10.1142/9789813228887_0022

Algebraic generating functions for Gegenbauer polynomials. / Maier, Robert S.

Frontiers In Orthogonal Polynomials and Q-series. Vol. 1 World Scientific Publishing Co. Pte Ltd, 2018. p. 425-444.

Research output: Chapter in Book/Report/Conference proceedingChapter

Maier, RS 2018, Algebraic generating functions for Gegenbauer polynomials. in Frontiers In Orthogonal Polynomials and Q-series. vol. 1, World Scientific Publishing Co. Pte Ltd, pp. 425-444. https://doi.org/10.1142/9789813228887_0022
Maier RS. Algebraic generating functions for Gegenbauer polynomials. In Frontiers In Orthogonal Polynomials and Q-series. Vol. 1. World Scientific Publishing Co. Pte Ltd. 2018. p. 425-444 https://doi.org/10.1142/9789813228887_0022
Maier, Robert S. / Algebraic generating functions for Gegenbauer polynomials. Frontiers In Orthogonal Polynomials and Q-series. Vol. 1 World Scientific Publishing Co. Pte Ltd, 2018. pp. 425-444
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