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

By means of Abel’s lemma on summation by parts, we derive several infinite series identities, which involve the classical harmonic numbers and their variants.

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).

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).

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.

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.

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.

In this paper we shall deduce a Bailey type formula as a consequence of the residual identity of a q q -series transformation due to Gasper. Our formula leads to a variety of q q -series identities which are related to the arithmetic function counting integer representations of the form \[ n ( A n + B ) 2 + r ( C r + D ) 2 + E n r . \frac {n(An+B)}{2}+\frac {r(Cr+D)}{2} + Enr. \]

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.