Euler-type sums involving odd harmonic numbers and binomial coefficients

Abstract

In this paper, by iterated integral expression of multiple t-polylogarithm function, we establish some expressions of series involving multiple t-harmonic sums in terms of coloured multiple zeta values. Using these expressions and the stuffle relations, we discuss the evaluations of some Euler-type sums involving odd harmonic numbers and binomial coefficients, such as T odd , k ( q ; r ) := ∑ n = 1 ∞ h n − 1 ( k 1 ) ⋯ h n − 1 ( k p ) ( 2 n − 1 ) q ( n + r r ) , T evev , k ( q ; r ) := ∑ n = 1 ∞ h n ( k 1 ) ⋯ h n ( k p ) n q ( n + r r ) , T odd , k ( q ; r ) := ∑ n = 1 ∞ h n − 1 ( k ) ( 2 n − 1 ) q ∏ i = 1 b ( n + r i r i ) , T even , k ( q ; r ) := ∑ n = 1 ∞ h n ( k ) n q ∏ i = 1 b ( n + r i r i ) , T odd , k 1 , k 2 ( q ; r ) := ∑ n = 1 ∞ h n − 1 ( k 1 ) h n − 1 ( k 2 ) ( 2 n − 1 ) q ∏ i = 1 b ( n + r i r i ) , T even , k 1 , k 2 ( q ; r ) := ∑ n = 1 ∞ h n ( k 1 ) h n ( k 2 ) n q ∏ i = 1 b ( n + r i r i ) , and some other forms. We present some explicit evaluations as examples. It can be found that this work gives a unified approach to such sums, and generalizes many known results in the literature.