SOME ORTHOGONAL VERY-WELL-POISED _{8}φ_{7}-FUNCTIONS THAT GENERALIZE ASKEY-WILSON POLYNOMIALS

SERGEI K. SUSLOV

Abstract. In a recent paper Ismail, Masson, and Suslov have established a continuous orthogonality relation and some other properties of a _{2}φ_{1}-Bessel function on a q-quadratic grid. Dick Askey suggested that the “Bessel-type orthogonality” at the _{2}φ_{1}-level has really a general character and can be extended up to the _{8}φ_{7}- level. Very-well-poised _{8}φ_{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}φ_{7}-functions. Another type of the orthogonality relation for a very-well-poised 8φ_{7}-function was recently found by Askey, Rahman, and Suslov.