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$q$-differential operator identities and applications. (English) Zbl 1114.33023
The author defines the $q$-exponential operator $$ _{1}\Phi_{0}\left(\aligned b \\ -\, \endaligned; q,\,-c\theta\right) =\sum_{n=0}^{\infty}\frac{(b;q)_{n}\,(-c\theta)^{n}}{(q;q)_{n}}\,, $$ and gives some of its properties such as transformation formulas and expressions in terms of basic hypergeometric series of $_{3}\Phi_{2}$ and $_{2}\Phi_{1}$ type. The author gives a $_{2}\Phi_{2}$ transformation formula which contains Jackson’s $_{2}\Phi_{2}$ transformation as a special case. The author also gives a formal extension of Bailey’s $_{3}\psi_{3}$ summation formula for bilateral basic hypergeometric series and describes a method of deriving a generalized formula of the Sears three terms of $_{3}\Phi_{2}$ series transformation. An extension of the Sears terminating balanced $_{4}\Phi_{3}$ transformation formula as well as a formal extension of Heine’s $_{2}\Phi_{1}$ transformation formula are also given.

33D15Basic hypergeometric functions of one variable, ${}_r\phi_s$
33D90Applications of basic hypergeometric functions
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A30$q$-calculus and related topics
Full Text: DOI
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