Sum of interpolated finite multiple harmonic q-series

Abstract

We define and study the interpolated finite multiple harmonic q-series. A generating function of the sums of the interpolated finite multiple harmonic qseries with fixed weight, depth and i-height is computed. Some Ohno-Zagier type relation with corollaries and some evaluation formulas of the interpolated finite multiple harmonic q-series at roots of unity are given.respectively. Here for any m ∈ N, [m] is the q-integer [m] = 1−q m 1−q . It was proved in [2, Theorems 1.1, 1.2] that the values z n (k; ζ n ) and z ⋆ n (k; ζ n ) closely relate to the finite and the symmetric multiple zeta (star) values, where ζ n is a fixed primitive n-th root of unity. Motivated by these observations, the authors of [2] gave a re-interpretation of *

A note on the sum of finite multiple harmonic $q$-series on $r\text{-}(r+1)$ indices

“…Proof. We get the result by the same way as in [4,Lemma 4.1]. Using Lemma 2.2 and (2.1), if u 1 = • • • = u r = 0, we have…”

“…Let r ∈ N be fixed. In [4], Z. Li and E. Pan considered the generating function of the sums of the interpolated finite multiple harmonic q-series with fixed weight, depth and 1-height, 2-height, . .…”

“…H. Bachmann, Y. Takeyama and K. Tasaka studied the special values of the finite multiple harmonic q-series at roots of unity in [1,2]. Z. Li and E. Pan introduced the interpolated finite multiple harmonic q-series and considered the generating function of the sums of the interpolated finite multiple harmonic q-series with fixed weight, depth and i-height in [4]. Based on the main result of [4], we study the sum of finite multiple harmonic q-series on r-(r + 1) indices at root of unity in this paper.…”

“…Z. Li and E. Pan introduced the interpolated finite multiple harmonic q-series and considered the generating function of the sums of the interpolated finite multiple harmonic q-series with fixed weight, depth and i-height in [4]. Based on the main result of [4], we study the sum of finite multiple harmonic q-series on r-(r + 1) indices at root of unity in this paper. Now let ζ n be a fixed primitive n-th root of unity.…”