{"id":{"repo_id":"york","oai_identifier":"oai:yorkspace.library.yorku.ca:10315/40626"},"canonical_url":"https://search.dev.ndltd.org/etd/york/oai:yorkspace.library.yorku.ca:10315/40626","repository":{"repo_id":"york","name":"York University","base_url":"https://yorkspace.library.yorku.ca/oai/request"},"display":{"title":"Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi","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.","abstract_html":"We introduce classes of Ramanujan-like series for <span class=\"etd-inline-math\">\\frac{1}{&pi;}</span>, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting <span class=\"etd-inline-math\">H<sub>n</sub> = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n}</span> denote the <span class=\"etd-inline-math\">n<sup>\\text{th}</sup></span> harmonic number: \\begin{align*} &amp; \\sum _{n=1}^{\\infty } \\frac{\\binom{2 n}{n}^2 H_n}{16^n(2 n - 1)} = \\frac{ 8 \\ln (2) - 4 }{\\pi}, \\\\ &amp; \\sum _{n=1}^{\\infty } \\frac{\\binom{2 n}{n}^2 H_n}{16^n(2 n-3)} = \\frac{120 \\ln (2)-68 }{27 \\pi}, \\\\ &amp; \\sum _{n = 1}^{\\infty } \\frac{ \\binom{2 n}{n}^2 H_n}{ 16^{ n} (2 n - 5)} = \\frac{10680 \\ln (2) -6508}{3375 \\pi }, \\\\ &amp; \\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 <span class=\"etd-inline-math\">\\frac{1}{&pi;}</span> containing harmonic numbers. We present a generalization of the recently discovered harmonic summation formula $<span class=\"etd-inline-math\">\\sum<sub>n=1</sub><sup>\\infty</sup> \\binom{2n}{n}<sup>2</sup> \\frac{H<sub>n</sub>}{32<sup>n</sup>} = \\frac{\\Gamma<sup>2</sup> \\left( \\frac{1}{4} \\right)}{4 \\sqrt{&pi;}} \\left( 1 - \\frac{4 \\ln(2)}{&pi;} \\right) </span>$ 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 $<span class=\"etd-inline-math\"> \\sum <sub>n=1</sub><sup>\\infty </sup> \\frac{ \\binom{2 n}{n}<sup>2</sup> H<sub>n</sub>}{32<sup>n</sup> (n + 1)} = 8-\\frac{2 \\Gamma<sup>2</sup> \\left(\\frac{1} {4}\\right)}{&pi; <sup>3/2</sup>}-\\frac{4 &pi; <sup>3/2</sup>+16 \\sqrt{&pi; } \\ln (2)}{\\Gamma<sup>2</sup> \\left(\\frac{1}{4}\\right)}. </span>$ We also introduce a class of harmonic summations for Catalan&#x27;s constant $G$ and $\\frac{1}{\\pi}$ such as the series $<span class=\"etd-inline-math\"> \\sum <sub>n=1</sub><sup>\\infty </sup> \\frac{ \\binom{2 n}{n}<sup>2</sup> H<sub>n</sub>}{ 16<sup>n</sup> (n+1)<sup>2</sup>} = 16+\\frac{32 G-64 \\ln (2)}{&pi; }-16 \\ln (2), </span>$ 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.","abstract_has_math":true,"creators":["Campbell, John Maxwell"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Zabrocki, Mike"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-12-14","date_published":"2022-12-14","updated_at":"2026-07-24T06:33:58Z","subjects":["Mathematics"],"languages":["en"],"rights":["Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10315/40626","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Zabrocki, Mike"]},{"key":"dc:creator","label":"Author","values":["Campbell, John Maxwell"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-12-14T16:21:56Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-12-14T16:21:56Z"]},{"key":"dc:date.issued","label":"Date","values":["2022-12-14"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10315/40626"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:title","label":"Title","values":["Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi"]}]}],"canonical_facts":{"dc:contributor.advisor":["Zabrocki, Mike"],"dc:creator":["Campbell, John Maxwell"],"dc:date.accessioned":["2022-12-14T16:21:56Z"],"dc:date.available":["2022-12-14T16:21:56Z"],"dc:date.issued":["2022-12-14"],"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:title":["Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi"],"dc:type":["Electronic Thesis or Dissertation"]},"updated_at":"2026-07-24T06:33:58Z"}