zbMATH — the first resource for mathematics

Quadratic transformations and Guillera’s formulas for \(1/\pi^2\). (English. Russian original) Zbl 1144.33002
Math. Notes 81, No. 3, 297-301 (2007); translation from Mat. Zametki 81, No. 3, 335-340 (2007).
The quadratic transformations for \({}_{2}F_{1} \) series, \[ {}_{2}F_{1} (a,b; 1+a-b; z) = (1-z)^{-a} \cdot \;{}_{2}F_{1} \left(\frac{1}{2} a, \frac{1}{2} + \frac{1}{2} a-b; 1+a-b; \frac{-4z}{(1-z)^2}\right) \] due to Gauss, and for \({}_{3}F_{2} \) series, \[ \begin{split}{}_{3}F_{2} (a,b,c; 1+a-b, 1+a-c; z) \\= (1-z)^{-a} \cdot {}_{3}F_{2} \left(\frac{1}{2} a, \frac{1}{2} + \frac{1}{2} a, 1+a-b-c; 1+a-b, 1+a-c; \frac{-4z}{(1-z)^2}\right)\end{split} \] due to Whipple are well-known.
In this paper, the author proves an analogue for \({}_{5}F_{4} \) series that involves multiple hypergeometric series. By utilizing this quadratic transformation formula and results of J. Guillera, formulas for \(\frac{1}{\pi^2} \) which are given by certain \({}_{5}F_{4} \) hypergeometric series, he derives two new formulas for \(\frac{1}{\pi^2}. \)
This idea seems likely to be used to derive more formulas for \(\frac{1}{\pi^2}\).

33C20 Generalized hypergeometric series, \({}_pF_q\)
33F05 Numerical approximation and evaluation of special functions
11Y55 Calculation of integer sequences
Full Text: DOI arXiv
[1] J. Guillera, ”Some binomial series obtained by the WZ-method,” Adv. in Appl. Math. 29(4), 599–603 (2002). · Zbl 1013.33010
[2] J. Guillera, ”About a new kind of Ramanujan-type series,” Experiment. Math. 12(4), 507–510 (2003). · Zbl 1083.33004
[3] J. Guillera, ”Generators of some Ramanujan formulas,” Ramanujan J. 11(1), 41–48 (2006). · Zbl 1109.33029
[4] J. Guillera, My Formulas for 1/{\(\pi\)} 2, Manuscript at http://personal.auna.com/jguillera/pi-formulas.pdf (2005).
[5] L. J. Slater, Generalized Hypergeometric Functions (Cambridge Univ. Press, Cambridge, 1966). · Zbl 0135.28101
[6] Y. Yang, ”On differential equations satisfied by modular forms,” Math. Z. 246(1–2), 1–19 (2004). · Zbl 1108.11040
[7] H. H. Chan, S. H. Chan, and Z. Liu, ”Domb’s numbers and Ramanujan-Sato type series for 1/{\(\pi\)},” Adv. Math. 186(2), 396–410 (2004). · Zbl 1122.11087
[8] Y. Yang (private communication, August 2005).
[9] G. Almkvist and W. Zudilin [V. V. Zudilin], ”Differential equations, mirror maps and zeta values,” in Mirror Symmetry V, AMS/IP Stud. Adv. Math., Ed. by N. Yui, S.-T. Yau, and J. D. Lewis (Amer. Math. Soc. & International Press, Providence, RI, 2007), Vol. 38, pp. 481–515; arXiv: math.NT/0402386. · Zbl 1118.14043
[10] W. Zudilin [V. V. Zudilin], ”Well-poised hypergeometric transformations of Euler-type multiple integrals,” J. London Math. Soc. (2) 70(1), 215–230 (2004). · Zbl 1065.11054
[11] M. Petkovšek, H. S. Wilf, and D. Zeilberger, A = B (A. K. Peters, Ltd., Wellesley, MA, 1996).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.