Related papers: On ${}_5\psi_5$ identities of Bailey

Let $F_k$ be the $k$th Fibonacci number. Let $(G_k)_{k\in\mathbb Z}$ be any sequence obeying the recurrence relation of the Fibonacci numbers. We employ the Gerin-Ces\`aro identity and an identity of Brousseau to evaluate the following…

Bhoria, Eyyunni and Maji recently obtained a four-parameter $q$-series identity which gives as special cases not only all five entries of Ramanujan on pages 354 and 355 of his second notebook but also allows them to obtain an analytical…

Euler's sum formula and its multi-variable and weighted generalizations form a large class of the identities of multiple zeta values. In this paper we prove a family of identities involving Bernoulli numbers and apply them to obtain…

Using a simple classical method we derive bilateral series identities from terminating ones. In particular, we show how to deduce Ramanujan's 1-psi-1 summation from the q-Pfaff-Saalschuetz summation. Further, we apply the same method to our…

We prove a new Bailey-type transformation relating WP-Bailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.

The $\mathrm{A}_2$ Bailey chain of Andrews, Schilling and the author is extended to a four-parameter $\mathrm{A}_2$ Bailey tree. As main application of this tree, we prove the Kanade-Russell conjecture for a three-parameter family of…

We prove a number of new Rogers-Ramanujan type identities involving double, triple and quadruple sums. They were discovered after an extensive search using Maple. The main idea of proofs is to reduce them to some known identities in the…

We obtain extensions of classical hypergeometric identities of Bailey and Whipple that transform nearly-poised and very-well-poised series to Saalsch\"utzian series, Saalsch\"utzian series to Saalsch\"utzian series, and very-well-poised and…

The notion of integral Bailey pairs is introduced. Using the single variable elliptic beta integral, we construct an infinite binary tree of identities for elliptic hypergeometric integrals. Two particular sequences of identities are…

Let ${\mathcal{P}_{n}}$ denote the set of positive integers which are prime to $n$. Let $B_{n}$ be the $n$-th Bernoulli number. For any prime $p\ge 5$ and $r\ge 2$, we prove that \begin{equation} \sum\limits_{\begin{smallmatrix}…

In this paper, we first establish two new Bailey pairs via finding two generalizations of Euler's pentagonal number theorem. Next, we specificize the Bailey lemmas with these two Bailey pairs. As applications, we finally establish some…

We show that identities involving trigonometric sums recently proved by Harshitha, Vasuki and Yathirajsharma, using Ramanujan's theory of theta functions, were either already in the literature or can be proved easily by adapting results…

We provide new proofs to five of Ramanujan's intriguing identities on false theta functions without using the Rogers-Fine identity and Bailey transforms.

In this paper, we continue our investigation of double sums where the inner sum is binomial but incomplete. We prove many new results for these types of double sums associated with binomial transform pairs. As applications we deduce new…

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)…

The classical summation and transformation theorems for very well-poised hypergeometric functions, namely, $_{5}F_4(1)$ summation, Dougall's $_{7}F_6(1)$ summation, Whipple's $_{7}F_6(1)$ to $_{4}F_3(1)$ transformation and Bailey's…

We derive two new identities involving the Bernoulli numbers, the Euler numbers, and the Stirling numbers of the first kind using analytic continuation of a well known identity for the Stirling numbers of the first kind.

We offer a more general Bailey pair than one that was proved in two different papers by two different methods [5, 12].

This paper highlights three known identities, each of which involves sums over alternating sign matrices. While proofs of all three are known, the only known derivations are as corollaries of difficult results. The simplicity and natural…

In this paper, by the method of comparing coefficients and the inverse technique, we establish the corresponding variate forms of two identities of Andrews and Yee for mock theta functions, as well as a few allied but unusual $q$-series…