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q-differential operator identities and applications. (English) Zbl 1114.33023

The author defines the q-exponential operator

1 Φ 0 b-;q,-cθ= n=0 (b;q) n (-cθ) n (q;q) n ,

and gives some of its properties such as transformation formulas and expressions in terms of basic hypergeometric series of 3 Φ 2 and 2 Φ 1 type. The author gives a 2 Φ 2 transformation formula which contains Jackson’s 2 Φ 2 transformation as a special case. The author also gives a formal extension of Bailey’s 3 ψ 3 summation formula for bilateral basic hypergeometric series and describes a method of deriving a generalized formula of the Sears three terms of 3 Φ 2 series transformation. An extension of the Sears terminating balanced 4 Φ 3 transformation formula as well as a formal extension of Heine’s 2 Φ 1 transformation formula are also given.


MSC:
33D15Basic hypergeometric functions of one variable, r φ s
33D90Applications of basic hypergeometric functions
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A30q-calculus and related topics