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

We introduce classes of Ramanujan-like series for \frac{1}{π}, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting H_n = 1 + \frac{1}{2} + \cdots + \frac{1}{n} denote the n^\text{th} 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 \frac{1}{π} 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}{32ⁿ} = \frac{\Gamma² \left( \frac{1}{4} \right)}{4 \sqrt{π}} \left( 1 - \frac{4 \ln(2)}{π} \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}{32ⁿ (n + 1)} = 8-\frac{2 \Gamma² \left(\frac{1} {4}\right)}{π ^3/2}-\frac{4 π ^3/2+16 \sqrt{π } \ln (2)}{\Gamma² \left(\frac{1}{4}\right)}. $ We also introduce a 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}{ 16ⁿ (n+1)²} = 16+\frac{32 G-64 \ln (2)}{π }-16 \ln (2), $ which we prove through a variation of our previous integration method for constructing $\frac{1}{\pi}$ 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 $\frac{1}{\pi^2}$, especially the constant $\frac{\zeta(3)}{\pi^2}$, which is of number-theoretic interest.

Author dc:creator

• Campbell, John Maxwell

Advisor dc:contributor.advisor

• Zabrocki, Mike

• Mathematics

Statement dc:rights

• Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.

Language dc:language

en

Handle dc:identifier.uri

http://hdl.handle.net/10315/40626

OAI identifier oai:identifier

oai:yorkspace.library.yorku.ca:10315/40626