Infinite Series Identities on Harmonic Numbers

Chapter 9 of Ramanujan’s Second Notebook: Infinite Series Identities, Transformations, and Evaluations

A general transformation formula between an infinite double series and an infinite single sum is established, which specializes to several infinite double series identities, including a recent one due to Lyons, Paule and Riese (2002).

By means of two hypergeometric summation formulae, we establish two large classes of infinite series identities with harmonic numbers and central binomial coefficients. Up to now, these numerous formulae have hidden behind very few known identities of Apéry-like series for Riemann-zeta function, discovered mainly by Lehmer [14] and Elsner [12] as well as Borwein et al. [4, 5, 7]. All the computation and verification are carried out by an appropriately-devised Mathematica package.

A variety of identities involving harmonic numbers and generalized harmonic numbers have been investigated since the distant past and involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics, and theoretical physics. Here we show how one can obtain further interesting and (almost) serendipitous identities about certain finite or infinite series involving binomial coefficients, harmonic numbers, and generalized harmonic numbers by simply applying the usual differential operator to well-known Gauss’s summation formula for2F1(1).

Eight infinite series involving harmonic-like numbers are coherently and systematically reviewed. They are evaluated in closed form exclusively by integration together with calculus and complex analysis. In particular, a mysterious series W is introduced and shown to be expressible in terms of the trilogarithm function. Several remarkable integral values and difficult infinite series identities are shown as consequences.

Binomial series involving harmonic polynomials are expressed in terms of parametric integrals. By evaluating these parametric integrals, we establish several remarkable closed formulae for the infinite series containing both central binomial coefficients and harmonic numbers. Most of the values for binomial series found in this paper concern the dilogarithm and trilogarithm functions.

The unified Ω-series of the Gauss and Bailey $$_2F_1(\frac{1}{2})$$-sums will be investigated by utilizing asymptotic methods and the modified Abel lemma on summation by parts. Several remarkable transformation theorems for the Ω-series will be proved whose particular cases turn out to be strange evaluations of nonterminating hypergeometric series and infinite series identities of Ramanujan-type, including a couple of beautiful expressions for π and the Catalan constant discovered by Guillera (2008).

B.C. Berndt has established many relations between various infinite series using a transformation formula for a large class of functions, which comes from a more general class of Eisenstein series. In this paper, continuing his study, we find some infinite series identities.

Applying the modified Abel lemma on summation by parts, we examine infinite series containing generalized harmonic numbers of order 2 and 3. Several interesting results mainly for $$\pi ^2$$ and $$\zeta (3)$$ are established as consequences.

By applying multiplicate forms of the Carlitz inverse series relations to the $q$-Pfaff-Saalschtz summation theorem, we establish twenty five nonterminating $q$-series identities with several of them serving as $q$-analogues of infinite series expressions for $\pi$ and $1/\pi$, including some typical ones discovered by Ramanujan (1914) and Guillera.

This contribution, which is a follow-up to author’s paper [1] dealing with the sums of the series of reciprocals of some quadratic polynomials, deals with the series of reciprocals of the cubic polynomials with triple non-positive integer root. Three formulas for the sum of this kind of series expressed by means of harmonic numbers are derived and presented, together with one approximate formula, and verified by several examples evaluated using the basic programming language of the computer algebra system Maple 16. This contribution can be an inspiration for teachers who are teaching the topic Infinite series or as a subject matter for work with talented students.

In this paper, we study the alternating Euler T -sums and S ˜ -sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler T -sums and S ˜ -sums, from which, the parity theorems of Hoffman’s double and triple t -values and Kaneko–Tsumura’s double and triple T -values are further obtained. As supplements, we also show that the linear T -sums and S ˜ -sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.