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Multiple q-zeta functions and multiple q-polylogarithms

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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 ℂd with explicit poles. In our Main Theorem we show that its limit when q 1 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 Jackson’s 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.

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Correspondence to Jianqiang Zhao.

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Partially supported by NSF grant DMS0139813 and DMS0348258.

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Zhao, J. Multiple q-zeta functions and multiple q-polylogarithms. Ramanujan J 14, 189–221 (2007). https://doi.org/10.1007/s11139-007-9025-9

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  • DOI: https://doi.org/10.1007/s11139-007-9025-9

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