Infinite series involving harmonic numbers and reciprocal of binomial coefficients

• Published in: AIMS mathematics Vol. 9; no. 7; pp. 16885 - 16900
• Main Authors: Chen, Kwang-Wu, Yang, Fu-Yao
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.05.2024
• Subjects: 2-poset; binomial coefficients; harmonic numbers; multiple zeta values; yamamoto's integral
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2024820

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: <disp-formula> <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> 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 $.