Extension of Hoffman’s Combinatorial Identity via Specific Zeta-Like Series

Abstract

The goal of the paper is twofold. First, we present an analytic method leading to a class of combinatorial identities with Bernoulli, Euler and Catalan numbers based on considering specific multiple zeta-like series and infinite products. The developed method allows us to naturally extend Hoffman’s combinatorial identity that led to the famous evaluation of the multiple zeta value ζ({2}k)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\zeta (\\{2\\}_k)$$\\end{document} in 1992. Second, we present new evaluations of two multiple zeta-like series with their consequences to combinatorial identities, and, as a by-product of our technical considerations, we establish two combinatorial identities with a trinomial coefficient and Stirling numbers respectively.