Summation formulae for twisted cubic q‐series

Abstract

A new class of twisted cubic q‐series is investigated by means of the modified Abel lemma on summation by parts. Several remarkable summation and transformation formulae are established for both terminating and nonterminating series.

Twisted cubic theta hypergeometric series

“…• The formulae presented in this paper are much deeper than their q-series counterparts appearing in Chu, 29 even though most of them have almost the same expressions except for the presence of the new "base" parameter "p." • For the nonterminating elliptic series, their convergence is even uncertain. However, there exist elegant identities for nonterminating q-series corresponding to Theorems 2.5, 2.12, and 2.13.…”

“…Whittaker and Watson 30, P451 and Rosengren 9 and Chu 31,32 ) ⟨a∕b, a∕c, a∕d, a∕e; p⟩ − ⟨b, c, d, e; p⟩ = b⟨a, a∕bc, a∕bd, a∕be; p⟩ (7) under the "well-poised" condition a 2 = bcde. Recently, the modified Abel lemma on summation by parts has successfully been employed by Chu [23][24][25][26][27][28][29]33 to review numerous transformation and summation formulae for basic hypergeometric series and by Chu-Jia [20][21][22] to investigate systematically theta series identities. In order to make the paper self-contained, this lemma is recorded as follows:…”

“…It should be pointed out that the two cubic series Ω n and n are elliptic analogues of the twisted q-series found recently by Chu. 29 The following observations may help to clear the differences around these two topics:…”

“…By means of the modified Abel lemma on summation by parts (cf. Chu [23][24][25][26][27][28][29] ), we shall establish 13 closed summation formulae for the Ω n -series in the next section (the central part of the paper). Then, the paper will end up with Section 3, where we shall briefly highlight the corresponding formulae for the reversal m -series.…”

“…Observe that the Ω n ‐series terminates, for , both below by the denominator parameters and above by the numerator parameters in the following 13 different cases: By means of the modified Abel lemma on summation by parts (cf. Chu 23‐29 ), we shall establish 13 closed summation formulae for the Ω n ‐series in the next section (the central part of the paper). Then, the paper will end up with Section 3, where we shall briefly highlight the corresponding formulae for the reversal ω m ‐series.…”