Some Basic Hypergeometric Orthogonal Polynomials That Generalize Jacobi Polynomials 1. Introduction. A hypergeometric series has the form V an with an+1 fan a rational function of n. A basic hypergeometric series has an+\/a„ a rational function of qn for a fixed q. The standard notation for hypergeometric series is M n F(au~"ar-x\-S% {ai)n '' *{ar)n*n (lA} r *\bu...,bs'x)-£ 0 (bi)n---(bs)nn\ where the shifted factorial (a)n is defined by -1), " = 1,2, n = 0. 1.2) (a)| | = ^ + l ) - - - ( a + if For basic hypergeometric series set —Q (b\\q)n- -(br+y, with )(\-aq). . .(l-aq*-*), n = 1,2,. ^(a 0 g) n ---(a r g) B (-iy n g Wjr* = 0 + i f (bx q)„-• •{br+j q)„{q q)„ ' J " - 1 ' " " (i.4) « «„-(J 1 ""* ( - * = o The series in (1.3) converges for all x when \q\ 1, j = 1,2,..., and no zeros appear in the denominator, and for |*| 1 wheny = 0. The case \q\ 1 can be transformed to \q\ 1 since (1.5) (a q)n = {-l)nanqv)(a-l q-l) n . Received by the editors November 16, 1981. Supported in part by NSF grant MCS 8101568.
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