A WZ PROOF OF RAMANUJAN’S FORMULA FOR π

Shalosh B. EKHAD^{1} and Doron ZEILBERGER^{1}

Dedicated to Archimedes on his 2300^{th} birthday

Archimedes computed π very accurately. Much later, Ramanujan discovered several infinite series for 1/π that enables one to compute π even more accurately. The most impressive one is([Ra]): ((a)_{k} denotes, as usual, a(a + 1)...(a + k − 1).)

\frac{1}{π} = 2\sqrt{2} ∑_{k=0}^{∞} \frac{(1/4)_{k}(1/2)_{k}(3/4)_{k}}{k!^{3}} (1103 + 26390k)(1/99)^{4k+2}.  (1)

This formula is an example of a non-terminating hypergeometric series identity. Many times, nonterminating series are either limiting cases or ”analytic continuations” of terminating identities, which are now known to be routinely provable by computer. [WZ].

While we do not know of a terminating generalization of (1), we do know how to give a WZ proof of another formula for π, also given by Ramanujan[Ra], and included in his famous letter to Hardy. This formula is:

\frac{2}{π} = ∑_{k=0}^{∞}(−1)^{k}(4k + 1)\frac{(1/2)_{k}^{3}}{k!^{3}} .  (2)

The terminating version, that we will prove is

\frac{Γ(3/2 + n)}{Γ(3/2)Γ(n + 1)} = ∑_{k=0}^{∞}(−1)^{k}(4k + 1)\frac{(1/2)_{k}^{2}(−n)_{k}}{k!^{2}(3/2 + n)_{k}} .  (3)

To prove it for all positive integers n, we call the summand divided by the left side F (n, k), and cleverly construct

G(n, k) := \frac{(2k + 1)^{2}}{(2n + 2k + 3)(4k + 1)}F (n, k) ,

with the motive that F (n + 1, k) − F (n, k) = G(n, k) − G(n, k − 1) (check!), and summing this last identity w.r.t k shows that ∑_{k} F (n, k) ≡ Constant, which is seen to be 1, by plugging in n = 0. This proves (3). To deduce (2), we ”plug” in n = −1/2, which is legitimate in view of Carlson’s theorem [Ba].

REFERENCES

[Ba] W.N. Bailey, “Generalized Hypergeometric Series”, (Cambridge Univ. Press, 1935), p. 39.

1 Department of Mathematics, Temple University, Philadelphia, PA 19122. [ekhad,zeilberg]@math.temple.edu ;

http://www.math.temple.edu/~[ekhad,zeilberg]. The work of the second author was supported in part by the NSF. This paper was published in p.107-108 of ‘Geometry, Analysis, and Mechanics’, ed. by J. M. Rassias, World Scientific, Singapore 1994.

[Ra] K.S. Rao, in “Srinivasa Ramanujan”, ed. K.R. Nagarajan and T. Soundararajan, (MACMIL-LAN INDIA, Madras, 1988).

[WZ] H. S. Wilf and D. Zeilberger, Amer. Math.Soc. B3 (1990) 147.