Hypergeometric (super)congruences

The sequence of (terminating balanced) hypergeometric sums an = n ∑ k=0 ( n k )2( n+ k k )2 , n = 0, 1, . . . , appears in Apéry’s proof of the irrationality of ζ(3). Another example of hypergeometric use in irrationality problems is Ramanujan-type identities for 1/π, like ∞ ∑ k=0 ( 2k k )3 (4k + 1) (−1) 26k = 2 π . These two, seemingly unrelated but both beautiful enough, hypergeometric series have many issues in common, as I explain in my review “Ramanujan-type formulae for 1/π. A second wind?”. In my talk, I plan to discuss further number-theoretical aspects of the two examples, namely, the congruences anp ≡ an (mod p) for n = 0, 1, . . . , p > 3 prime (I. Gessel, 1982), and p−1 ∑ k=0 ( 2k k )3 (4k + 1) (−1) 26k ≡ (−1)(p−1)/2p (mod p) for p > 2 prime (E. Mortenson, 2008); these are called supercongruences because they happen to hold not just modulo a prime p but a higher power of p. In spite of the elementary character of these supercongruences, the existing proofs are not general enough to treat other similar cases. My goal is to attract attention to this nice and elementary subject on the border of arithmetic and hypergeometrics. 1. Hypergeometric series The (generalized) hypergeometric function is defined by the series mFm−1 ( a1, a2, . . . , am b2, . . . , bm ∣∣∣∣ z) = ∞ ∑ n=0 (a1)n(a2)n · · · (am)n (b2)n · · · (bm)n z n! , (1) which has the unit disc |z| < 1 as natural domain of convergence, where (a)n = Γ(a+ n) Γ(a) = { a(a+ 1) · · · (a+ n− 1) if n ≥ 1, 1 if n = 0, Date: July 20–25, 2009. 1