Symmetry techniques for $q$-series: Askey-Wilson polynomials

We advocate the exploitation of symmetry (recurrence relation) techniques for the derivation of properties associated with families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. Here these ideas are applied to the (continuous) Askey-Wilson polynomials, introduced by Askey and Wilson, to obtain a strikingly simple derivation of their orthogonality relations. 1. In t roduct ion . In [1] the authors and A.K. Agarwal introduced symmetry techniques for the study of families of basic hypergeometric functions, in analogy with the local Lie theory techniques for ordinary hypergeometric functions. The fundamental objects in this study are the recurrence relations obeyed by the families: generating functions and identities for each family are characterized in terms of the recurrence relations. The functions studied in [1] are of the form ^ / a i , . . . , a r \ _ y > (ai;q)k'"(ar;q)k z k V 6 i , . . . , 6 8 ' ' / f^Q(bi;q)k'"(b8;q)k{q;q)k and many-variable extension. Here , . _ J 1 if k = 0 ( f l ; g , ' " \ ( l a ) ( l g a ) . . . ( l c * 1 o ) if fc = 1,2,... and \q\ . ( , , . ) ( . ( I ^ ) ( I ^ " E.G. KALNINS AND W. MILLER, JR. 225 Here Eff(z) = f(qz). Relation (2.1A) follows from T(az;q)J-;q) = J ^ ( l q)(aqz;q)k.1 (^— ;q) \Z /k q' \ Z / k-l and (2.IB) follows from KaZ;q)k(±q)k = (1 a b q ^ ^ q ) ^ ^ q ) ^ The first relation was pointed out by Askey and Wilson [2]; we have not found (2. IB) in the literature. Consider relation (2.IB); it suggests that there should be an operator Aaq-^,bq-^,cq,dq) . Aa,b,c,d) T , , a A l± mapping Cn t o