Related papers: Double series for $\pi$ and their $q$-analogues

We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence $\{g(k)\}$), to be reduced to an infinite $q$-product times a single basic…

Recursive formulas extending some known $_{2}F_{1}$ and $_{3}F_{2}$ summation formulas by using contiguous relations have been obtained. On the one hand, these recursive equations are quite suitable for symbolic and numerical evaluation by…

Using $q$-series identities and series rearrangement, we establish several extensions of $q$-Watson formulas with two extra integer parameters. Then they and Sears' transformation formula are utilized to derive some generalizations of…

The classical hypergeometric summation theorems are exploited to derive several striking identities on harmonic numbers including those discovered recently by Paule and Schneider (2003).

This survey article (which will appear as a chapter in the book ``Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions'', Springer-Verlag) provides a small collection of basic material on multiple…

In terms of the analytic continuation method, we prove three transformation formulas involving bilateral basic hypergeometric series. One of them is equivalent to Jouhet's result involving two $_8\psi_8$ series and two $_8\phi_7$ series.

Several new $q$-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the…

We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic…

We give the hypergeometric solutions of some algebraic equations including the general fifth degree equation.

Two integral representations of q-analogues of the Hurwitz zeta function are established. Each integral representation allows us to obtain an analytic continuation including also a full description of poles and special values at…

According to the method of series rearrangement, we establish two families of summation formulae for $q$-Watson type $_4\phi_3$-series.

We prove several infinite families of $q$-series identities for false theta functions and related series. These identities are motivated by considerations of characters of modules of vertex operator superalgebras and of quantum…

We use one-dimensional double affine Hecke algebras to introduce q-counterparts of the Gauss integrals and new types of Gauss-Selberg sums at roots of unity.

Two integral solutions of q-difference equations of the hypergeometric type with |q|=1 are constructed by using the double sine function. One is an integral of the Barnes type and the other is of the Euler type.

We prove a conjecture that arose in the context of a subspace enumeration problem over finite fields. We prove, more generally, a bibasic, double-sum identity, which extends a $q$-analogue of the (terminating) binomial theorem.

We construct a family of bilinear differential operators which satisfy certain gauge properties. These operators can be naturally associated with $q$-deformations of classical integrable hierarchies. In particular, we consider the case when…

We establish several summation formulae for hypergeometric and basic hypergeometric series involving noncommutative parameters and argument. These results were inspired by a recent paper of J. A. Tirao [Proc. Nat. Acad. Sci. 100 (14)…

By using contiguous relations for basic hypergeometric series, we give simple proofs of Bailey's $_4\phi_3$ summation, Carlitz's $_5\phi_4$ summation, Sears' $_3\phi_2$ to $_5\phi_4$ transformation, Sears' ${}_4\phi_3$ transformations,…

A doubly infinite set of series expansion for $1/\pi$ are reported. They follow trivially from a formal expansion for the quotient of the values taken by the gamma function for two (complex) arguments differing by an integer plus one half,…

With the help of a summation of basic hypergeometric series, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials, we find some new $q$-supercongruences.…