A q-microscope for supercongruences

Abstract

By examining asymptotic behavior of certain infinite basic (q-) hypergeometric sums at roots of unity (that is, at a ‘q-microscopic’ level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a q-analogue of Ramanujan's formula∑n=0∞(4n2n)(2nn)228n32n(8n+1)=23π, of the two supercongruencesS(p−1)≡p(−3p)(modp3)andS(p−12)≡p(−3p)(modp3), valid for all primes p>3, where S(N) denotes the truncation of the infinite sum at the N-th place and (−3⋅) stands for the quadratic character modulo 3.