More Ramanujan-type formulae for \(1/\pi^2\).

*(English. Russian original)*Zbl 1203.11088
Russ. Math. Surv. 62, No. 3, 634-636 (2007); translation from Usp. Mat. Nauk 62, No. 3, 211-212 (2007).

From the text: Some of the most spectacular achievements in the history of the number \(\pi\) are the representations of \(1/\pi\) by rapidly converging series discovered by S. Ramanujan in 1914 [Q. J. Math., Oxf. (2) 45, 350–372 (1914; JFM 45.1249.01), see also Collected papers of Srinivasa Ramanujan, Cambridge Univ. Press, Cambridge (1927; JFM 53.0030.02), pp. 23–39]. Although Ramanujan himself did not explain how he arrived at his series, he indicated that they belong to what is now known as ‘the theories of elliptic functions to alternative bases’.
The first rigorous mathematical proofs of Ramanujan’s identities in [loc. cit.] and generalizations of them were given by J. M. Borwein and P. B. Borwein [Pi and the AGM, New York etc.: John Wiley (1987; Zbl 0611.10001] and D. V. Chudnovsky and G. V. Chudnovsky [Ramanujan revisited, Proc. Conf., Urbana-Champaign/Illinois 1987, 375–472 (1988; Zbl 0647.10002)]. One of the now standard examples of Ramanujan-type formulae is the Chudnovskys’ famous formula

\[ \sum_{n=0}^{\infty} \frac{\big(\frac{1}{6}\big)_{n}\big(\frac{1}{2}\big)_{n}\big(\frac{5}{6}\big)_ {n}}{n!^3} (545140134n+13591409)\cdot \frac{(-1)^{n}}{53360^{3n}} = \frac{3\cdot 53360^ 2}{2\pi \sqrt{10005}}, \]

which enabled them to hold the record for the calculation of \(\pi\) in 1989–1994. Here \((a)_n =\Gamma(a+n)/\Gamma(a) = a(a+1) \cdots (a+n-1)\) for \(n > 1\) and \((a)_0 = 1\) is the Pochhammer symbol.

Quite recently, following a different method, J. Guillera [Adv. Appl. Math. 29, No. 4, 599–603 (2002; Zbl 1013.33010), Ramanujan J. 11, No. 1, 41–48 (2006; Zbl 1109.33029)] managed not only to prove some of Ramanujan’s identities but also to derive similar rapidly converging series for \(1/\pi^2\). In this note we present a simple algorithm for producing Ramanujan–Guillera-type formulae for \(1/\pi^2\) and the more complicated representations for the higher powers of \(\frac{1}{\pi}\) from known ones for \(1/\pi\).

Finally several examples of the Ramanujan representations of the type discussed here are presented.

\[ \sum_{n=0}^{\infty} \frac{\big(\frac{1}{6}\big)_{n}\big(\frac{1}{2}\big)_{n}\big(\frac{5}{6}\big)_ {n}}{n!^3} (545140134n+13591409)\cdot \frac{(-1)^{n}}{53360^{3n}} = \frac{3\cdot 53360^ 2}{2\pi \sqrt{10005}}, \]

which enabled them to hold the record for the calculation of \(\pi\) in 1989–1994. Here \((a)_n =\Gamma(a+n)/\Gamma(a) = a(a+1) \cdots (a+n-1)\) for \(n > 1\) and \((a)_0 = 1\) is the Pochhammer symbol.

Quite recently, following a different method, J. Guillera [Adv. Appl. Math. 29, No. 4, 599–603 (2002; Zbl 1013.33010), Ramanujan J. 11, No. 1, 41–48 (2006; Zbl 1109.33029)] managed not only to prove some of Ramanujan’s identities but also to derive similar rapidly converging series for \(1/\pi^2\). In this note we present a simple algorithm for producing Ramanujan–Guillera-type formulae for \(1/\pi^2\) and the more complicated representations for the higher powers of \(\frac{1}{\pi}\) from known ones for \(1/\pi\).

Finally several examples of the Ramanujan representations of the type discussed here are presented.

Reviewer: Olaf Ninnemann (Berlin)

##### MSC:

11Y60 | Evaluation of number-theoretic constants |