Q-integral Representation Research Articles

Multiple q-zeta values

We introduce a q-analog of the multiple harmonic series commonly referred to as multiple zeta values. The multiple q-zeta values satisfy a q-stuffle multiplication rule analogous to the stuffle multiplication rule arising from the series representation of ordinary multiple zeta values. Additionally, multiple q-zeta values can be viewed as special values of the multiple q-polylogarithm, which admits a multiple Jackson q-integral representation whose limiting case is the Drinfel'd simplex integral for the ordinary multiple polylogarithm when q = 1 . The multiple Jackson q-integral representation for multiple q-zeta values leads to a second multiplication rule satisfied by them, referred to as a q-shuffle. Despite this, it appears that many numerical relations satisfied by ordinary multiple zeta values have no interesting q-extension. For example, a suitable q-analog of Broadhurst's formula for ζ ( { 3 , 1 } n ) , if one exists, is likely to be rather complicated. Nevertheless, we show that a number of infinite classes of relations, including Hoffman's partition identities, Ohno's cyclic sum identities, Granville's sum formula, Euler's convolution formula, Ohno's generalized duality relation, and the derivation relations of Ihara and Kaneko extend to multiple q-zeta values.

ON q-ANALOG OF HUMMER'S THEOREM AND ITS CONTIGUOUS RESULTS

The aim of this paper is to derive the well-known q-analog of kummer's theorem by using q-integral representation. In addition to this, two results closely related to the q-kummer's theorem have also been obtained by the same method.

(Shifted) Macdonald Polynomials: q-Integral Representation and Combinatorial Formula

We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We strengthen some theorems of F. Knop and S. Sahi and give two explicit formulas for these polynomials: a q-integral representation and a combinatorial formula. Our main tool is a q-integral representation for ordinary Macdonald polynomial. We also discuss duality for shifted Macdonald polynomials and Jack degeneration of these polynomials.

Q-integral representations of modified q-Bessel functions and q-Macdonald functions

The q-analogues of modified Bessel functions and Macdonald functions were defined in the previous paper of the authors as general solutions of certain second-order difference equations. Several representations of these functions based on the Jackson integral are presented.

Some Summation Theorems and Transformations for Q-Series

AbstractWe introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the 6ϕ5 summation formula of F. H. Jackson and the q-Dixon sum. We also give q-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found.

A q-integral representation of Rogers' q-ultraspherical polynomials and some applications

Aq-integral representation of Rogers'q-ultraspherical polynomialsC n (x;β∥q) is obtained by using Sears' summation formula for balanced non-terminating3 φ 2 series. It is then used to give a simple derivation of the Gasper-Rahman formula for the Poisson kernel ofC n (x;β∥q). As another application it is shown how this representation can be directly used to give an asymptotic expansion of theq-ultraspherical polynomials.