Related papers: Transformations and summations for bilateral basic…
In this paper we consider a transformation $L_a$ of sequences of complex numbers. We find the inverse transformation of $L_a$ as well as the inverse of a related transformation $\tilde{L}_a$. We explore a connection to the binomial…
By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8\phi_7$-series. Their duplicate forms under the partition…
Our object is a thorough analysis of the WP-Bailey tree, a recent extension of classical Bailey chains. We begin by observing how the WP-Bailey tree naturally entails a finite number of classical q-hypergeometric transformation formulas. We…
The double hypergeometric Kamp\'e de F\'eriet series $F^{0:3}_{1:1}(1,1)$ depends upon 9 complex parameters. We present three cases with 2 relations between those 9 parameters, and show that under these circumstances $F^{0:3}_{1:1}(1,1)$…
A multiple generalization of elliptic hypergeometric series is investigated and a duality transformation for multiple hypergeometric series is proposed. Our duality transformation obtained from an identity arising from the Cauchy…
We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram…
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…
In a prior paper we found that the Fourier-Legendre series of a Bessel function of the first kind J_{N}\left(kx\right) and of a modified Bessel functions of the first kind I_{N}\left(kx\right) lead to an infinite set of series involving…
Let $F(n,k)$ be a hypergeometric function that may be expressed so that $n$ appears within initial arguments of inverted Pochhammer symbols, as in factors of the form $\frac{1}{(n)_{k}}$. Only in exceptional cases is $F(n, k)$ such that…
In this paper we construct a discrete linear operator $K$ which transforms $A_2$ Macdonald polynomials into the product of two basic $3\phi_2$ hypergeometric series with known arguments. The action of the operator $K$ on power sums in two…
We prove a master theorem for hypergeometric functions of Karlsson-Minton type, stating that a very general multilateral U(n) Karlsson-Minton type hypergeometric series may be reduced to a finite sum. This identity contains the…
By using two known transformation formulas for basic hypergeometric series, we establish a direct extension of Bailey's $_6\psi_6$-series identity. Subsequently, it and Milne's identity are employed to drive multi-variable generalizations…
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 introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic…
Recently, Bringmann and Kane established two new Bailey pairs and used them to relate certain q-hypergeometric series to real quadratic fields. We show how these pairs give rise to new mock theta functions in the form of q-hypergeometric…
A new multiple-integral representation of a general family of very-well-poised hypergeometric series is proved. Inspite of an analytic character of the result, it is motivated by the recent arithmetic progress for the values of the Riemann…
We give a method to embed the q-series in a (p,q)-series and derive the corresponding (p,q)-extensions of the known q-identities. The (p,q)-hypergeometric series, or twin-basic hypergeometric series (diferent from the usual bibasic…
We present a systematic method for proving nonterminating basic hypergeometric identities. Assume that $k$ is the summation index. By setting a parameter $x$ to $xq^n$, we may find a recurrence relation of the summation by using the…
In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric…
In recent years, combinatorial reconfiguration problems have attracted great attention due to their connection to various topics such as optimization, counting, enumeration, or sampling. One of the most intriguing open questions concerns…