On q-analogs of Multiple Zeta Values and other Multiple Harmonic Series.

Abstract. An example of a multiple harmonic sum is Z_n(s,t,u) := S_n>k>j>m>0 k^-s j^-t m^-u. Here, the sum is over all positive integers k, j, m satisfying the indicated inequalities, which in some cases may be weak instead of strict as shown. The bound n is either a positive integer or infinite. The variables s, t, u are unrestricted if n is finite, but are usually assumed to be positive integers, with s>1 if n is infinite to ensure convergence. In general, we may have an arbitrary finite number of variables instead of three. The q-analog of a positive integer k is [k]_q := S_j=0^k-1q^j = (1-q^k)/(1-q), 0 < q < 1. Naively, one can obtain a q-analog of multiple harmonic sums by replacing the summation indices by their respective q-analogs. However, this approach needs to be modified in order to yield interesting results. I shall discuss what sorts of results can be obtained with appropriate modifications, outline a few of the techniques used to prove them, and hopefully give an indication of why researchers are interested in this subject.