New series involving harmonic numbers and squared central binomial coefficients

John Campbell (1)

1 Department of Mathematics and Statistics [Toronto]

Recently, there have 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 $$\sum_{n=1}^{\infty} \binom{2n}{n}^{2} \frac{H_{n}}{32^{n}} = \frac{\Gamma^{2} \left( \frac{1}{4} \right)}{4 \sqrt{\pi}} \left( 1 - \frac{4 \ln(2)}{\pi} \right) $$ 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 $$ \sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{32^n (n + 1)} = 8-\frac{2 \Gamma \left(\frac{1} {4}\right)^2}{\pi ^{3/2}}-\frac{4 \pi ^{3/2}+16 \sqrt{\pi } \ln (2)}{\Gamma \left(\frac{1}{4}\right)^2} $$ 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.

Mots clés: Harmonic numbers; Central binomial coefficient; Infinite series; Symbolic computation; Gamma function

Domaines: Mathématiques [math]; Analyse classique [math.CA]

John Campbell. New series involving harmonic numbers and squared central binomial coefficients. 2018. ⟨hal-01774708⟩