Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi
| dc.contributor.advisor | Zabrocki, Mike | |
| dc.contributor.author | Campbell, John Maxwell | |
| dc.date.accessioned | 2022-12-14T16:21:56Z | |
| dc.date.available | 2022-12-14T16:21:56Z | |
| dc.date.copyright | 2022-04-20 | |
| dc.date.issued | 2022-12-14 | |
| dc.date.updated | 2022-12-14T16:21:56Z | |
| dc.degree.discipline | Mathematics & Statistics | |
| dc.degree.level | Doctoral | |
| dc.degree.name | PhD - Doctor of Philosophy | |
| dc.description.abstract | We introduce classes of Ramanujan-like series for $\frac{1}{\pi}$, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting $H_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ 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 $\frac{1}{\pi}$ 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 $\frac{1}{\pi}$ 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. 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. | |
| dc.identifier.uri | http://hdl.handle.net/10315/40626 | |
| dc.language | en | |
| dc.rights | Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests. | |
| dc.subject | Mathematics | |
| dc.subject.keywords | Infinite series | |
| dc.subject.keywords | Harmonic number | |
| dc.subject.keywords | Pi formula | |
| dc.subject.keywords | Elliptic integral | |
| dc.subject.keywords | Binomial coefficient | |
| dc.subject.keywords | Symbolic computation | |
| dc.subject.keywords | Gamma function | |
| dc.subject.keywords | Integral transform | |
| dc.subject.keywords | Hypergeometric series | |
| dc.subject.keywords | Riemann zeta function | |
| dc.subject.keywords | Legendre polynomial | |
| dc.title | Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi | |
| dc.type | Electronic Thesis or Dissertation |
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