Some new formulas involving \(\Gamma_q\) functions
Summary: In a recent paper [Rend. Semin. Mat. Univ. Padova 112, 199--235 (2004; Zbl 1167.33308)] we found some new results for \(q\)-functions of many variables with the aid of the \(\Gamma_q\) function. The Heine notation reminding of the hypergeometric case was used throughout, and some relations between \(\Gamma_q\) functions were presented. This paper aims at giving the promised longer exposition of \(\Gamma_q\) -- revealing also the connection between this and the Jacobi-theta functions which appear in context. We will give a slightly generalized definition of the Heine series with more general tilde operators. Four \(q\)-summation formulas of Andrews will be given in the new notation. The close affinity to \(q\)-binomial coefficient formulas will be stressed by expressing the finite \(q\)-hypergeometric formulas, the canonical form, in two ways. Two further \(q\)-analogues of Kummer's \(_2F_1(-1)\) formula will be given. An ancient \(q\)-analogue of the Euler reflection formula will be used for the proof of a special case of the Bailey-Daum summation formula, conjectured in the previous paper. Multiple extensions of Gauss' formula will be given by a similar technique. All this will explain the utility of the Heine notation.
scientific article; zbMATH DE number 5376051
scientific article