Infinite series involving harmonic numbers and reciprocal of binomial coefficients

Abstract

<abstract><p>Yamamoto's integral was the integral associated with 2-posets, which was first introduced by Yamamoto. In this paper, we obtained the values of infinite series involving harmonic numbers and reciprocal of binomial coefficients by using some techniques of Yamamoto's integral. We determine the value of infinite series of the form:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \sum\limits_{m_1,\ldots,m_n,\ell_1,\ldots,\ell_k\geq 1}\frac{H_{m_1}^{(a_1)}\cdots H_{m_n}^{(a_n)}} {m_1^{b_1}\cdots m_n^{b_n}\ell_1^{c_1}\cdots\ell_k^{c_k} \binom{m_1+\cdots+m_n+\ell_1+\cdots+\ell_k}{\ell_k}}, $\end{document} </tex-math></disp-formula></p> <p>in terms of a finite sum of multiple zeta values, for positive integers $ a_1, \ldots, a_n, b_1, \ldots, b_n, c_1, \ldots, c_k $.</p></abstract>

On Some General Tornheim-Type Series

“…This approach elegantly leads to our results. A similar method was used in [26] by the author and Yang to derive infinite series involving harmonic numbers and the reciprocals of binomial coefficients.…”