### 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 language | English (US) |
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Title of host publication | Frontiers In Orthogonal Polynomials and Q-series |

Publisher | World Scientific Publishing Co. Pte Ltd |

Pages | 425-444 |

Number of pages | 20 |

Volume | 1 |

ISBN (Electronic) | 9789813228887 |

ISBN (Print) | 9789813228870 |

DOIs | |

State | Published - Jan 12 2018 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

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## 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