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Abstract. By means of the Carlitz ^-analogue of Gould-Hsu inverse relations, a quick introduction to the basic hypergeometric formulae is presented in an elementary and direct way. As a unified treatment, several famous identities, e.g., the ^-analogues of the DougallDixon-Watson-Whipple formulas, are shown to be the dual versions of the ^-Saalschutz theorem. 1991 Mathematics Subject Classification: 05A30; 33A30. 0. Introduction Following the usual notation, the basic hypergeometric series with base q and variable z (where \q\ < l and \z\ < 1) is defined by (cf. e.g., [12]) with <7-shifted factorials (O.lb) (*; q)a = ft (l - xq)n = (x; q)J(xq"; ?),, k =0 and ) ( 01 1 (gt; q\(a2\ q\ ... (flr; q)n instead of factorial-fractions. Partially supported by Alexander von Humboldt Foundation (Germany) Italian Consiglio Nazionale Delle Ricerche (CNR) NSF (Chinese), Youth-Grant 1990-1033 ChuWenchang After sleeping many years in the mathematical world, the basic hypergermetric series woke up and has aroused new interest during the last decade for its wide applications to mathematics, physics and Computer science. But most relations about ^-series seem to be like monsters to the non-specialist outside the circle of special functions. This fact has stimulated the author to seek
Forum Mathematicum – de Gruyter
Published: Jan 1, 1995
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