Some Orthogonal Very-well-poised 8'7-functions That Generalize Askey{wilson Polynomials

In a recent paper Ismail et al. (Algebraic Methods and q-Special Functions (J.F. van Diejen and L. Vinet, eds.) CRM Proceding 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. Verywell-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.