### Abstract

In a recent paper Ismail et al. (Algebraic Methods and q-Special Functions (J.F. van Diejen and L. Vinet, eds.) CRM Preceding and Lecture Notes, Vol. 22. American Mathematical Society, 1999, pp. 183-200) have established a continuous orthogonality relation and some other properties of a 2φ1-Bessel function on a q-quadratic grid. Dick Askey (private communication) suggested that the "Bessel-type orthogonality" found in Ismail et al. (1999) at the 2φ1-level has really a general character and can be extended up to the 8φ7-level. Very-well-poised 8φ7-functions are known as a nonterminating version of the classical Askey-Wilson polynomials (SIAM J. Math. Anal. 10 (1979), 1008-1016; Memoirs Amer. Math. Soc. Number 319 (1985)). Askey's conjecture has been proved by the author in J. Phys. A: Math. Gen. 30 (1997), 5877-5885. In the present paper which is an extended version of Suslov (1997) we discuss in detail properties of the orthogonal 8φ7-functions. Another type of the orthogonality relation for a very-well-poised 8φ7-function was recently found by Askey et al. J. Comp. Appl. Math. 68 (1996), 25-55.

Original language | English (US) |
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Pages (from-to) | 183-218 |

Number of pages | 36 |

Journal | Ramanujan Journal |

Volume | 5 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2001 |

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### Keywords

- Askey-Wilson polynomials
- Basic hypergeometric series
- Orthogonal functions
- q-Bessel functions

### ASJC Scopus subject areas

- Algebra and Number Theory