NEW SERIES INVOLVING HARMONIC NUMBERS AND SQUARED CENTRAL BINOMIAL COEFFICIENTS

Recently, there has been a variety of intriguing discoveries regarding the symbolic computation of series containing central binomial coefficients and harmonic-type numbers. In this article, we present a vast generalization of the recently discovered harmonic summation formula Sigma(infinity)(n=1) ((2n)(n))(2) H-n/32(n) = Gamma(2)(1/4)/4 root pi (1- 4 ln 2/pi) through creative applications of an integration method that we had previously introduced and applied to prove new Ramanujan-like formulas for 1/pi. We provide explicit closed-form expressions for natural variants of the above series that cannot be evaluated by state-of-the-art computer algebra systems, such as the elegant symbolic evaluation Sigma(infinity)(n=1) ((2n)(n))(2) H-n/32(n)(n + 1) = 8 - 2 Gamma(2)(1/4)/pi(3/2) - 4 pi(3/2) + 16 root pi ln 2/Gamma(2)(1/4) introduced in our present paper. We also discuss some related problems concerning binomial series containing alternating harmonic numbers. We also introduce a new class of harmonic summations for Catalan's constant G and 1/pi such as the series Sigma(infinity)(n=1) ((2n)(n))(2) H-n/16(n)(n + 1)(2) = 16 + 32G - 64 ln 2/pi - 16 ln 2 which we prove through a variation of our previous integration method for constructing 1/pi series.