ABSTRACT

In 1981, Andrews gave a four-variable generalization of Ramanujan's ${}_1{\psi }_1$ summation formula. We establish a six-variable generalization of Andrews' identity according to the transformation formula for two ${}_8{\varphi }_7$ series and Bailey's transformation formula for three ${}_8{\varphi }_7$ series. Then, it is used to find a six-variable generalization of Ramanujan's reciprocity theorem, which is different from Liu's formula. We derive the generalizations of Bailey's two ${}_3{\psi }_3$ summation formulas in terms of two limiting relations and Bailey's another transformation formula for three ${}_8{\varphi }_7$ series. Based on the two limiting relations, some different results involving bilateral basic hypergeometric series are also deduced from the Guo–Schlosser transformation formula and other two transformation formulas.

摘要

1981年，安德鲁斯（Andrews）对拉马努詹（Ramanujan）的四变量进行了概括 ${}_{\mathrm{1个}}{\psi }_{\mathrm{1个}}$求和公式。根据两个的转换公式，我们建立了一个六变量的安德鲁斯身份的推广。${}_8{\varphi }_7$ 级数和贝利的三个变换公式 ${}_8{\varphi }_7$系列。然后，它被用来寻找Ramanujan互易定理的六变量泛化，这与Liu的公式不同。我们推导了Bailey的两个泛化${}_3{\psi }_3$ 两个极限关系的求和公式，以及贝利针对三个极限关系的另一个变换公式 ${}_8{\varphi }_7$系列。基于这两个限制关系，还从Guo-Schlosser转换公式和其他两个转换公式推导出了涉及双边基本超几何级数的一些不同结果。