Abstract
Abstract In this paper, we mainly show that generalized Euler-type sums of multiple harmonic sums with reciprocal binomial coefficients can be expressed in terms of rational linear combinations of products of classical multiple zeta values (MZVs) and multiple harmonic star sums (MHSSs). Furthermore, applying the stuffle relations, we prove that the Euler-type sums involving products of generalized harmonic numbers and reciprocal binomial coefficients can be evaluated by MZVs and MHSSs.
Highlights
We mainly show that generalized multiple harmonic sums (MHSs) with reciprocal binomial coefficients of types
From [8, Theorem 3.1], we know that the key idea to study the following more general Euler-type sums
Using (2.5) and (3.3) yields the desired formula
Summary
For a composition k = (k1,...,kr) and positive integer n, the classical multiple harmonic sums (MHSs) and the classical multiple harmonic star sums (MHSSs) are defined by. They proved that the sums Sπ1,q(k) and Sπq1(k) can be expressed as rational linear combinations of products of zeta values, linear Euler sums (or double zeta values), harmonic numbers and double harmonic star sums. Can be expressed in terms of linear combinations of classical MHSs and classical MZVs with depth less than or equal to r. Applying the stuffle relations, called quasi-shuffle relations (see [11]), we know that for any composition k = (k1,...,kr), the product Hn(k1) ⋯ Hn(kr) can be expressed in terms of linear combinations of MHSs (for the explicit formula, see [12, equation (2.4)]).
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