Ramanujan’s Theories of Elliptic Functions to Alternative Bases

Bruce C. Berndt
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Bruce C. Berndt: University of Illinois at Urbana-Champaign, Department of Mathematics

Chapter 33 in Ramanujan’s Notebooks, 1998, pp 89-181 from Springer

Abstract: Abstract In his famous paper [3], [10, pp. 23–39], Ramanujan offers several beautiful series representations for 1/pi. He first states three formulas, one of which is $$ \frac{4}{\pi } = \sum\limits_{{\mathbf{n = 0}}}^\infty {\frac{{\left( {6{\mathbf{n}} + 1} \right){{\left( {\frac{1}{2}} \right)}^3}_n}}{{{{\left( {{\mathbf{n}}!} \right)}^3}{4^n}}}} $$ where (a)o = 1 and, for each positive integer n $$ {\left( {\mathbf{a}} \right)_n} = {\mathbf{a}}\left( {{\mathbf{a}} + 1} \right)\left( {{\mathbf{a}} + 2} \right)...\left( {{\mathbf{a}} + {\mathbf{n}} - 1} \right) $$ .

Keywords: Modular Form; Elliptic Function; Eisenstein Series; Part Versus; Alternative Basis (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1007/978-1-4612-1624-7_3

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