the electronic journal of combinatorics 5 (1998), #R10

A BINOMIAL COEFFICIENT IDENTITY

ASSOCIATED TO A CONJECTURE OF BEUKERS

Scott Ahlgren, Shalosh B. Ekhad, Ken Ono and Doron Zeilberger

Abstract. Using the WZ method, a binomial coefficient identity is proved. This identity is noteworthy since its truth is known to imply a conjecture of Beukers.

Received: January 28, 1998; Accepted: February 1, 1998

If n is a positive integer, then let A(n) := ∑_{k=0}^{n} \binom{n}{k}^{2}\binom{n + k}{k}^{2}, and define integers a(n) by

∑_{n=1}^{∞} a(n)q^{n} := q ∏_{n=1}^{∞}(1 − q^{2n})^{4}(1 − q^{4n})^{4} = q − 4q^{3} − 2q^{5} + 24q^{7} − · · · .

Beukers conjectured that if p is an odd prime, then

(1)  A (\frac{p − 1}{2}) ≡ a(p) (mod p^{2}).

In [A-O] it is shown that (1) is implied by the truth of the following identity.

Theorem. If n is a positive integer, then

∑_{k=1}^{n} k\binom{n}{k}^{2}\binom{n + k}{k}^{2} \{\frac{1}{2k} + ∑_{i=1}^{n+k} \frac{1}{i} + ∑_{i=1}^{n−k} \frac{1}{i} − 2 ∑_{i=1}^{k} \frac{1}{i}\} = 0.

Remark. This identity is easily verified using the WZ method, in a generalized form [Z] that applies when the summand is a hypergeometric term times a WZ potential function. It holds for all positive n, since it holds for n=1,2,3 (check!), and since the sequence defined by the sum satisfies a certain (homog.) third order linear recurrence equation. To find the recurrence, and its proof, download the Maple package EKHAD and the Maple program zeilWZP from http://www.math.temple.edu/~ zeilberg . Calling the quantity inside the braces c(n, k), compute the WZ pair (F, G), where F = c(n, k + 1) − c(n, k) and G = c(n + 1, k) − c(n, k).

Go into Maple, and type read zeilWZP; zeilWZP(k*(n+k)!**2/k!**4/(n-k)!**2,F,G,k,n,N):

References

[A-O] S. Ahlgren and K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences (in prepa-

ration).

[B] F. Beukers, Another congruence for Apéry numbers, J. Number Th. 25 (1987), 201-210.

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

Department of Mathematics, Penn State University, University Park, Pennsylvania 16802 E-mail address: ahlgren@math.psu.edu

Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122 E-mail address: ekhad@math.temple.edu; http://www.math.temple.edu/ ˜ ekhad

Department of Mathematics, Penn State University, University Park, Pennsylvania 16802 E-mail address: ono@math.psu.edu; http://www.math.psu.edu/ono/

Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122 E-mail address: zeilberg@math.temple.edu; http://www.math.temple.edu/˜ zeilberg

The third author is supported by NSF grant DMS-9508976 and NSA grant MSPR-Y012. The last author is supported in part by the NSF.

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