New series involving harmonic numbers and squared central binomial coefficients

Abstract

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 $$\sum _{n=1}^{\infty } \binom {2n}{n}^{2} \frac {H_{n}}{32^{n}} = \frac {\Gamma^2\bigl(\frac {1}{4} )}{4 \sqrt {\pi }} ( 1 - \frac {4 \ln 2}{\pi } ) $$ through creative applications of an integration method that we had previously introduced and applied to prove new Ramanujan-like formulas for $\frac {1}{\pi }$. 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 $$ \sum _{n=1}^{\infty } \frac { \binom {2 n}{n}^2 H_n}{32^n (n + 1)} = 8-\frac {2 \Gamma^2\bigl(\frac {1} {4} )}{\pi ^{3/2}}-\frac {4 \pi ^{3/2}+16 \sqrt {\pi } \ln 2}{\Gamma^2\bigl(\frac {1}{4})} $$ 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 $\frac {1}{\pi }$ such as the series $$ \sum _{n=1}^{\infty } \frac { \binom {2 n}{n}^2 H_n}{ 16^{n} (n+1)^2} = 16+\frac {32 G-64 \ln 2}{\pi }-16 \ln 2 $$ which we prove through a variation of our previous integration method for constructing $\frac {1}{\pi }$ series.