Summation, Transformation, and Expansion Formulas for Bibasic Series

Abstract

Abstract. An indefinite bibasic sum containing three parameters is evaluated and used to derive bibasic extensions of Euler's transformation formula and of a Fields and Wimp expansion formula. It is also used to derive a transformation formula involving four independent bases, a 9-Lagrange inversion formula, and some quadratic, cubic and quartic summation formulas.

Essays on the theory of elliptic hypergeometric functions

“…In conclusion of this section, let us mention that the quadratic transformations for the s+1 V s -series and related summation formulae were considered in [49,52,53,54]. Other specific elliptic hypergeometric series were investigated in the papers [55,56]. An interesting application of a s+1 E s -series with the nontrivial power variable y appeared recently in [57].…”

Очерки Теории Эллиптических Гипергеометрических Функций

“…В заключение этого раздела упомянем, что квадратичные преобразования для s+1 V s -рядов и связанные формулы суммирования рассмотрены в [49], [52]- [54]. Другие специфические эллиптические гипергеометрические ряды ис-следовались в работах [55], [56]. Интересное приложение s+1 E s -рядов с нетри-виальным степенным параметром y появилось недавно в [57].…”

Partial Sums of Two Quartic q-Series

“…In the theory of basic hypergeometric series, there are several important classes, for example, well-poised [3], quadratic [13,17], cubic [16] and quartic [10,11] series. To our knowledge, there are four quartic series which can be displayed as follows: G n (a, c, e) := …”

“…By means of the series rearrangement, Gasper and Rahman [10,11,12] discovered several summation and transformation formulae for the nonterminating special cases of F n (a, b, d) and U n (a, b, d). The terminating series identities for the last four sums have been established by Chu [4] and Chu-Wang [8], respectively, through inversion techniques and Abel's lemma on summation by parts.…”