The Australian Journal of Mathematical Analysis and Applications AJMAA Volume 16, Issue 1, Article 14, pp. 1-14, 2019 CUBIC ALTERNATING HARMONIC NUMBER SUMS ANTHONY SOFO Received 12 September, 2018; accepted 23 January, 2019; published 27 May, 2019. VICTORIA UNIVERSITY,COLLEGE OF ENGINEERING AND SCIENCE,MELBOURNE CITY,AUSTRALIA.
[email protected] ABSTRACT. We develop new closed form representations of sums of cubic alternating harmonic numbers and reciprocal binomial coefficients. We also identify a new integral representation for the ζ (4) constant. Key words and phrases: Polylogarithm function, Alternating cubic harmonic numbers, Combinatorial series identities, Sum- mation formulas, Partial fraction approach, Binomial coefficients. 2000 Mathematics Subject Classification. Primary 05A19, 33C20. Secondary 11M06, 11B83. ISSN (electronic): 1449-5910 c 2019 Austral Internet Publishing. All rights reserved. 2 ANTHONY SOFO 1. INTRODUCTION AND PRELIMINARIES In this paper we will develop identities, closed form representations of alternating cubic har- monic numbers and reciprocal binomial coefficients, including integral representations, of the form: ∞ n+1 X (−1) H3 (1.1) Ω(k, p) = n , n + k n =1 np k th for p = 0, 1 and k ∈ N0. Here, the n harmonic number n X 1 Z 1 1 − tn (1.2) H = = γ + ψ (n + 1) = dt, H := 0 n r 1 − t 0 r=1 0 and as usual, γ denotes the Euler-Mascheroni constant and ψ(z) is the Psi (or Digamma) func- tion defined by d Γ0(z) Z z ψ(z) := {log Γ(z)} = or log Γ(z) = ψ(t) dt. dz Γ(z) 1 For sums of harmonic numbers with positive terms [10], [27], [28] and [29] have given many results, including sums of the form ∞ X H3 n .