THE RAMANUJAN JOURNAL, Vol. 14, No. 2 (2007),
Multiple $q$-zeta functions and multiple $q$-polylogarithms
Received September 16, 2003; Accepted October 25, 2006
Abstract. For every positive integer $d$ we define the $q$-analog of multiple zeta function of depth $d$ and study its properties, generalizing the work of Kaneko et al. who dealt with the case $d=1$. We first analytically continue it to a meromorphic function on $\C^d$ with explicit poles. In our Main Theorem we show that its limit when $q\uparrow1$ is the ordinary multiple zeta function. Then we consider some special values of these functions when $d=2$. At the end of the paper we also propose the $q$-analogs of multiple polylogarithms by using Jacksons $q$-iterated integrals and then study some of their properties. Our definition is motivated by those of Koornwinder and Schlesinger although theirs are slightly different from ours.
Keywords: Multiple $q$-zeta functions, multiple $q$-polylogarithms, shuffle relations, iterated integrals
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Updated Sat Sep 22 15:01:30 EDT 2007.