Related papers: The Abel Lemma and the q-Gosper Algorithm
Recently, Chen, Hou and Jin used both Abel's lemma on summation by parts and Zeilberger's algorithm to generate recurrence relations for definite summations. Meanwhile, they proposed the Abel-Gosper method to evaluate some indefinite sums…
The partial sums of two quartic basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several summation and transformation formulae are consequently established.
An elementary proof is given for a nonterminating "strange" cubic $_7F_6$-series summation formula of Gasper and Rahman, through the modified Abel lemma on summation by parts. As a byproduct, an interesting nonterminating…
In this paper, we establish three new and general transformations with sixteen parameters and bases via Abel's lemma on summation by parts. As applications, we set up a lot of new transformations of basic hypergeometric series. Among…
In this paper we introduce the so-called truncated very-well-poised $_6\psi_6$ series and set up an explicit recurrence relation for it by means of the classical Abel lemma on summation by parts. This new recurrence relation implies an…
We prove some new semi-finite forms of bilateral basic hypergeometric series. One of them yields in a direct limit Bailey's celebrated ${}_6\psi_6$ summation formula, answering a question recently raised by Chen and Fu ({\em Semi-Finite…
Computers are good at evaluating finite sums in closed form, but there are finite sums which do not have closed forms. Summands which do not produce a closed form can often be ``fixed'' by multiplying them by a suitable polynomial. We…
We use both Abel's lemma on summation by parts and Zeilberger's algorithm to find recurrence relations for definite summations. The role of Abel's lemma can be extended to the case of linear difference operators with polynomial…
We give summation formulae for the bilateral basic hypergeometric series ${}_1\psi_1( a; b; q, z )$ through Ramanujan's summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian…
We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised 6-psi-6 summation theorem, and…
We give new proofs for certain bilateral basic hypergeometric summation formulas using the symmetries of the corresponding series. In particular, we present a proof for Bailey's $_3\psi_3$ summation formula as an application. We also prove…
We show that several classical bilateral summation and transformation formulas have semi-finite forms. We obtain these semi-finite forms from unilateral summation and transformation formulas. Our method can be applied to derive Ramanujan's…
We derive two Gosper-type Lambert series identities of level $14$ which involve the $q$-constant $\Pi_q$ using a special case of Bailey's $_6\psi_6$ summation formula and certain propeties of $\eta$-quotients and generalized…
The Abramov-Petkovsek reduction computes an additive decomposition of a hypergeometric term, which extends the functionality of the Gosper algorithm for indefinite hypergeometric summation. We modify the Abramov-Petkovsek reduction so as to…
A multidimensional generalization of Bailey's very-well-poised bilateral basic hypergeometric ${}_6\psi_6$ summation formula and its Dougall type ${}_5H_5$ hypergeometric degeneration for $q\to 1$ is studied. The multiple Bailey sum amounts…
In the present paper, we establish three special $q$-Abel transformation formulae of $q$-series via the use of Abel's lemma on summation by parts. As direct applications, we set up the corresponding $q$-contiguous relations for three kinds…
We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which…
In a recent letter, new representations were proposed for the pair of sequences ($\gamma,\delta$), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs ($\gamma,\delta$) labelled by the…
By telescoping method, Sun gave some hypergeometric series whose sums are related to $\pi$ recently. We investigate these series from the point of view of Gosper's algorithm. Given a hypergeometric term $t_k$, we consider the Gosper…
We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauss' 2-F-1 summation and elementary series manipulations to give a simple proof of Dougall's 2-H-2…