Related papers: A Telescoping method for Double Summations

The aim of this research paper is to obtain explicit expressions of (i) $ {}_1F_1 \left[\begin{array}{c} \alpha \\ 2\alpha + i \end{array} ; x \right]. {}_1F_1\left[ \begin{array}{c} \beta \\ 2\beta + j \end{array} ; x \right]$ (ii)…

Explicit expressions for the hypergeometric series ${}_2F_1(-n, a; 2a\pm j;2)$ and ${}_2F_1(-n, a; -2n\pm j;2)$ for positive integer $n$ and arbitrary integer $j$ are obtained with the help of generalizations of Kummer's second and third…

We extend Petkov\v{s}ek's algorithm for computing hypergeometric solutions of scalar difference equations to the case of difference systems $\tau(Y) = M Y$, with $M \in {\rm GL}_n(C(x))$, where $\tau$ is the shift operator. Hypergeometric…

In this note, we aim to provide generalizations of (i) Knuth's old sum (or Reed Dawson identity) and (ii) Riordan's identity using a hypergeometric series approach.

The aim of this research paper is to demonstrate how one can obtain eleven new and interesting hypergeometric identities (in the form of a single result) from the old ones by mainly applying the well known beta integral method which was…

For a hypergeometric series $\sum_k f(k,a, b, ...,c)$ with parameters $a, b, >...,c$, Paule has found a variation of Zeilberger's algorithm to establish recurrence relations involving shifts on the parameters. We consider a more general…

We derive by analytic means a number of bilateral identities of the Rogers--Ramanujan type. Our results include bilateral extensions of the Rogers--Ramanujan and the G\"ollnitz-Gordon identities, and of related identities by Ramanujan,…

It is only in exceptional cases that a $_2F_1(z)$-series with rational parameters and a rational argument, apart from the cases for $z \in \{ \pm 1, \frac{1}{2} \}$ associated with classical hypergeometric identities, admits an evaluation…

The generalized hypergeometric function $_qF_p$ is a power series in which the ratio of successive terms is a rational function of the summation index. The Gaussian hypergeometric functions $_2F_1$ and $_3F_2$ are most common special cases…

We present a constructive proof of Jacobi's identity for the sum of two squares. We present a combinatorial proof of the Jacobi Triple Product and combine with a proof of Hirschhorn to define an algorithm. The input is a factorization…

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…

This paper shows that certain $\,_{3}F_{4}$ hypergeometric functions can be expanded in sums of pair products of $\,_{1}F_{2}$ functions. In special cases, the $\,_{3}F_{4}$ hypergeometric functions reduce to $\,_{2}F_{3}$ functions.…

We use Andrews' $q$-analogues of Watson's and Whipple's $_3F_2$ summation theorems to deduce two formulas for products of specific basic hypergeometric functions. These constitute $q$-analogues of corresponding product formulas for ordinary…

In this paper we review and derive hyperbolic and trigonometric double summation addition theorems for Jacobi functions of the first and second kind. In connection with these addition theorems, we perform a full analysis of the relation…

Let $\Delta_x f(x,y)=f(x+1,y)-f(x,y)$ and $\Delta_y f(x,y)=f(x,y+1)-f(x,y)$ be the difference operators with respect to $x$ and $y$. A rational function $f(x,y)$ is called summable if there exist rational functions $g(x,y)$ and $h(x,y)$…

We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $BC_n$ root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in…

We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler's Telescoping Lemma, we give alternate…

This work was intended to be all about, and only about, hypergeometric 3F2(1). The initial goal was to revisit many identities from the literature that have been derived over the years and show that they can be obtained in a simpler way…

We establish two binomial coefficient--generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of…

We give a simple geometric argument to derive in a common manner orthospectrum identities of Basmajian and Bridgeman. Our method also considerably simplifies the determination of the summands in these identities. For example, for every odd…