Generators of some Ramanujan formulas. (English) Zbl 1109.33029

Summary: We prove some Ramanujan type formulas for \(1/\pi\) but without using the theory of modular forms. Instead we use the WZ–method created by H. S. Wilf and D. Zeilberger [J. Am. Math. Soc. 3, No. 1, 147–158 (1990; Zbl 0695.05004)] and find some hypergeometric functions in two variables which are second components of WZ-pairs than can be certified using Zeilberger’s EKHAD package. These certificates have an additional property which allows us to get generalized Ramanujan’s type series which are routinely proven by computer. We call these second hypergeometric components of the WZ-pairs generators. Finding generators seems a hard task but using a kind of experimental research (explained below), we have succeeded in finding some of them. Unfortunately we have not found yet generators for the most impressive Ramanujan’s formulas. We also prove some interesting binomial sums for the constant \(1/\pi^{2}\). Finally we rewrite many of the obtained series using Pochhammer symbols and study the rate of convergence.


33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)


Zbl 0695.05004
Full Text: DOI arXiv


[1] Bailey, W.N.: Generalized hypergeometric series. Cambridge Univ. Press (1935), p. 39 · Zbl 0011.02303
[2] Ekhad, S.B., Zeilberger, D.: A WZ proof of Ramanujan’s formula for {\(\pi\)}. In: Rassias, J.M. (ed.). Geometry, Analysis and Mechanics, World Scientific, Singapore (1994) · Zbl 0849.33003
[3] Guillera, J.: Some binomial series obtained by the WZ-method. Adv. in Appl. Math. 29, 599–603 (2002) · Zbl 1013.33010
[4] Petkovšek, M., Wilf, H.S., Zeilberger, D.: A = B, Peters A.K.: Ltd., Appendix A (1996)
[5] Ramanujan, S.: Modular equations and approximations to {\(\pi\)}. Quart. J. Pure Appl. Math. 45, 350–372 (1914) · JFM 45.1249.01
[6] Wilf, H.S., Zeilberger, D.: Rational functions certify combinatorial identities. J. Amer. Math. Soc. 3, 147–158 (1990) · Zbl 0695.05004
[7] Zeilberger, D.: Closed-form (pun intended!). Contemp. Math. 143, 579–607 (1993) · Zbl 0808.05010
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.