\(q\)-differential operator identities and applications
The author defines the \(q\)-exponential operator \[ _{1}\Phi_{0}\left(\begin{aligned} b \\ -\, \end{aligned}; q,\,-c\theta\right) =\sum_{n=0}^{\infty}\frac{(b;q)_{n}\,(-c\theta)^{n}}{(q;q)_{n}}\,, \] 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.
scientific article; zbMATH DE number 5161783