Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi

We introduce classes of Ramanujan-like series for $1\pi$, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting $Hn=1+12+\cdots +1n$ denote the $nth$ harmonic number: \begin{align*} & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n - 1)} = \frac{ 8 \ln (2) - 4 }{\pi}, \ & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n-3)} = \frac{120 \ln (2)-68 }{27 \pi}, \ & \sum _{n = 1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{ n} (2 n - 5)} = \frac{10680 \ln (2) -6508}{3375 \pi }, \ & \cdots \end{align*} In this direction, our main technique is based on the evaluation of a parameter derivative of a beta-type integral, but we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for $1\pi$ containing harmonic numbers.

We present a 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. We provide explicit closed-form expressions for natural variants of the above series. At the time of our research being conducted, up-to-date versions of Computer Algebra Systems such as Mathematica and Maple could not evaluate our introduced series, such as $$ \sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{32^n (n + 1)} = 8-\frac{2 \Gamma^2 \left(\frac{1} {4}\right)}{\pi ^{3/2}}-\frac{4 \pi ^{3/2}+16 \sqrt{\pi } \ln (2)}{\Gamma^2 \left(\frac{1}{4}\right)}. $$ We also introduce a class of harmonic summations for Catalan's constant $G$ and $1\pi$ 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 $1\pi$ series.

We also present a new integration method for evaluating infinite series involving alternating harmonic numbers, and we apply a Fourier--Legendre-based technique recently introduced by Campbell et al., to prove new rational double hypergeometric series formulas for expressions involving $1\pi 2$, especially the constant $\zeta (3)\pi 2$, which is of number-theoretic interest.