Abel’s lemma on summation by parts and partial q-series transformations

Abstract

The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q-series are established.

Summation formulae for twisted cubic q‐series

“…Hence, the series Θ n ( a , b , c , d ) is self‐reciprocal in this sense. Under the replacement k → n − k on the summation index, we can check without difficulty that they satisfy the following reciprocal relation: Our approach will be the modified Abel lemma on summation by parts (cf Chu et al). In order to make the paper self‐contained, we record it as follows.…”

“…For both the terminating and nonterminating forms of the series (1), there was an intensive investigation made by Gasper, 7 Gasper-Rahman, 2( §3.8);8 and Rahman, 9,10 as well as the author and his collaborators. [11][12][13][14] The aim of this paper is to investigate the following "twisted cubic series":…”

“…Our approach will be the modified Abel lemma on summation by parts (cf Chu et al [12][13][14][15][16][17] ). In order to make the paper self-contained, we record it as follows.…”

“…Define the series Θ n ( a , b , c , d ) by the partial sum This series is called “cubic series” because the shifted factorials of order k appearing in the numerator and the denominator can be paired off so that for each pair, the product of their bases is q 4 and the product of the corresponding parameters is equal to “ a ” apart from a small integer power of the base “ q .” For both the terminating and nonterminating forms of the series , there was an intensive investigation made by Gasper, Gasper‐Rahman,(§3.8); and Rahman, as well as the author and his collaborators …”

Partial Sums of Two Quartic q-Series

“…[5,6,7,8]) as follows. For an arbitrary complex sequence {τ k }, define the backward and forward difference operators and ∆, respectively, by…”