### 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) |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*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

}

TY - CHAP

T1 - Algebraic generating functions for Gegenbauer polynomials

AU - Maier, Robert S

PY - 2018/1/12

Y1 - 2018/1/12

N2 - 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.

AB - 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.

KW - Gegenbauer polynomial

KW - Generating function

KW - Hypergeometric transformation

KW - Legendre function

KW - Legendre polynomial

KW - Poisson kernel

UR - http://www.scopus.com/inward/record.url?scp=85045706335&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85045706335&partnerID=8YFLogxK

U2 - 10.1142/9789813228887_0022

DO - 10.1142/9789813228887_0022

M3 - Chapter

AN - SCOPUS:85045706335

SN - 9789813228870

VL - 1

SP - 425

EP - 444

BT - Frontiers In Orthogonal Polynomials and Q-series

PB - World Scientific Publishing Co. Pte Ltd

ER -