FASTER AND FASTER CONVERGENT SERIES FOR ζ(3)

Tewodros Amdeberhan

Department of Mathematics, Temple University, Philadelphia PA 19122, USA tewodros@euclid.math.temple.edu

Submitted: April 8, 1996. Accepted: April 15, 1996

Abstract. Using WZ pairs we present accelerated series for computing ζ(3)

AMS Subject Classification: Primary 05A

Alf van der Poorten [P] gave a delightful account of Apéry’s proof [A] of the irrationality of ζ(3). Using WZ forms, that came from [WZ1], Doron Zeilberger [Z] embedded it in a conceptual framework.

We recall [Z] that a discrete function A(n,k) is called Hypergeometric (or Closed Form (CF)) in two variables when the ratios A(n + 1, k)/A(n, k) and A(n, k + 1)/A(n, k) are both rational functions. A pair (F,G) of CF functions is a WZ pair if F (n + 1, k) − F (n, k) = G(n, k + 1) − G(n, k). In this paper, after choosing a particular F (where its companion G is then produced by the amazing Maple package EKHAD accompanying [PWZ]), we will give a list of accelerated series calculating ζ(3). Our choice of F is

F (n, k) = \frac{(−1)^{k}k!^{2}(sn − k − 1)!}{(sn + k + 1)!(k + 1)}

where s may take the values s=1,2,3, … [AZ] (the section pertaining to this can be found in http://www.math.temple.edu/˜ tewodros). In order to arrive at the desired series we apply the following result:

Theorem: ([Z], Theorem 7, p.596) For any WZ pair (F,G)

∑_{n=0}^{∞} G(n, 0) = ∑_{n=1}^{∞} (F (n, n − 1) + G(n − 1, n − 1)) ,

whenever either side converges.

The case s=1 is Apéry’s celeberated sum [P] (see also [Z]):

ζ(3) = \frac{5}{2} ∑_{n=1}^{∞}(−1)^{n−1} \frac{1}{\binom{2n}{n}n^{3}}

where the corresponding G is

G(n, k) = \frac{2(−1)^{k}k!^{2}(n − k)!}{(n + k + 1)!(n + 1)^{2}} .

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For s=2 we obtain

ζ(3) = \frac{1}{4} ∑_{n=1}^{∞}(−1)^{n−1} \frac{56n^{2} − 32n + 5}{(2n − 1)^{2}} \frac{1}{\binom{3n}{n}\binom{2n}{n}n^{3}}

where G is

G(n, k) = \frac{(−1)^{k}k!^{2}(2n − k)!(3 + 4n)(4n^{2} + 6n + k + 3)}{2(2n + k + 2)!(n + 1)^{2}(2n + 1)^{2}}.

For s=3 we have

ζ(3) = ∑_{n=0}^{∞} \frac{(−1)^{n}}{72\binom{4n}{n}\binom{3n}{n}}\{\frac{6120n + 5265n^{4} + 13761n^{2} + 13878n^{3} + 1040}{(4n + 1)(4n + 3)(n + 1)(3n + 1)^{2}(3n + 2)^{2}}\},

and so on.

References

[A] R. Apéry, Irrationalitè de ζ(2) et ζ(3), Asterisque 61 (1979), 11-13.

[AZ] T. Amdeberhan, D. Zeilberger, WZ-Magic, in preparation.

[PWZ] M. Petkovšek, H.S. Wilf, D.Zeilberger, “A=B”, A.K. Peters Ltd., 1996.

The package EKHAD is available by the www at http://www.math.temple.edu/˜ zeilberg/programs.html

[P] A. van der Poorten, A proof that Euler missed ..., Apéry’s proof of the irrationality of ζ(3), Math. Intel. 1 (1979), 195-203.

[WZ1] H.S. Wilf, D. Zeilberger, Rational functions certify combinatorial identities, Jour. Amer. Math. Soc. 3 (1990), 147-158.

[Z] D. Zeilberger, Closed Form (pun intended!), Contemporary Mathematics 143 (1993), 579-607..