Abstract
We show how a couple of Ramanujan’s series for \(1/\pi \) can be deduced directly from Forsyth’s series and from Wallis’s product formula for \(\pi \). The same method is used to obtain Bauer’s alternating series.
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Bauer, G.: Von den Coefficienten der Reihen von Kugelfunctionen einer Variablen. J. Reine Angew. Math. 56, 101–121 (1859)
Ben-Ari, I., Hay, D., Roitershtein, A.: On Wallis-type products and Pólya’s urn schemes. Am. Math. Mon. 121(5), 422–432 (2014)
Baruah, N.D., Berndt, B.C., Chan, H.H.: Ramanujan’s series for \(1/\pi \): a survey. Am. Math. Mon. 116(7), 567–587 (2009)
Chu, W.: Dougall’s bilateral \( _2H_2\)-series and Ramanujan-like \(\pi \)-formulae. Math. Comp. 80, 2223–2251 (2011)
Dougall, J.: On Vandermonde’s theorem and some more general expansions. Proc. Edinb. Math. Soc. 25, 114–132 (1907)
Forsyth, A.R.: A Series for \(\frac{1}{\pi }\). Messenger Math. 12, 142–143 (1883)
Glaisher, J.W.L.: On series for \(\frac{1}{\pi }\) and \(\frac{1}{\pi ^2}\). Q. J. Pure Appl. Math. 37, 173–198 (1905–1906)
Guillera, J.: Series de Ramanujan: Generalizaciones y conjeturas. Ph.D. Thesis, University of Zaragoza, Spain (2007)
Guillera, J.: Accelerating Dougall’s \({}_5F_4\) sum and the WZ-algorithm. arXiv:1611.04385 (2016). Accessed 14 Mar 2017
Knopp, K.: Theory and Application of Infinite Series, 2nd edn. Blackie, London (1954). (4th reprint)
Levrie, P.: Using Fourier-Legendre expansions to derive series for \(1/\pi \) and \(1/\pi ^2\). Ramanujan J. 22(2), 221–230 (2010)
Liu, Z.-G.: A summation formula and Ramanujan type series. J. Math. Anal. Appl. 389(2), 1059–1065 (2012)
Liu, Z.-G.: Gauss summation and Ramanujan-type series for \(1/\pi \). Int. J. Number Theory 8(2), 289–297 (2012)
Nimbran, A.S.: Generalized Wallis-Euler products and new infinite products for \(\pi \). Math. Stud. 83(1–4), 155–164 (2014)
Nimbran, A.S.: Deriving Forsyth-Glaisher type series for \(\frac{1}{\pi }\) and Catalan’s constant by an elementary method. Math. Stud. 84(1–2), 69–86 (2015)
Ramanujan, S.: Modular equations and approximations to \(\frac{1}{\pi }\). Q. J. Pure Appl. Math. 45, 350–372 (1914). http://ramanujan.sirinudi.org/Volumes/published/ram06.pdf
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Levrie, P., Nimbran, A.S. From Wallis and Forsyth to Ramanujan. Ramanujan J 47, 533–545 (2018). https://doi.org/10.1007/s11139-017-9940-3
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DOI: https://doi.org/10.1007/s11139-017-9940-3