THE RAMANUJAN JOURNAL, Vol. 14, No. 2 (2007), 189-221.

Multiple $q$-zeta functions and multiple $q$-polylogarithms

JIANQIANG ZHAO
Department of Mathematics, Eckerd College, St. Petersburg, FL 33711, USA
Email: zhaoj@eckerd.edu
Homepage: http://www.math.upenn.edu/~jqz

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

2000 Mathematics Classification: Primary 11M41; 81R50; Secondary 11B68, 05A30, 11R42