Some orthogonal very-well-poised $_8\varphi_7$-functions that generalize Askey-Wilson polynomials

Abstract

In a recent paper Ismail, Masson, and Suslov have established a continuous orthogonality relation and some other properties of a $_2\varphi_1$-Bessel function on a $q$-quadratic grid. Dick Askey suggested that the ``Bessel-type orthogonality'' at the $_2\varphi_1$-level has really a general character and can be extended up to the $_8\varphi_7$-level. Very-well-poised $_8\varphi_7$-functions are known as a nonterminating version of the classical Askey--Wilson polynomials. Askey's congecture has been proved by the author. In the present paper we discuss in details some properties of the orthogonal $_8\varphi_7$-functions. Another type of the orthogonality relation for a very-well-poised $_8\varphi_7$-function was recently found by Askey, Rahman, and Suslov.