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 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.

A Common q-Analogue of Two Supercongruences

“…Note that Φ n (q)Φ n (−q) = Φ n (q 2 ) for odd indices n. The n ≡ 3 (mod 4) case of Theorem 1.1 confirms a conjecture of these authors [12,Conjecture 4.13], which states that, for n ≡ 3 (mod 4),…”

“…Like the proofs given in [12], we start with the following generalization of (1.7) with an extra parameter a. (…”

q-Analogues of Dwork-type supercongruences

“…We first give the following result, which follows from the c = 1 case of [15,Theorem 4.2]. Lemma 2.1.…”

“…The author [5] himself proved that the qsupercongruence (1.6) is true modulo [n]Φ n (q). Shortly afterwards, the author and Zudilin [15] proved that (1.6) holds modulo [n]Φ n (q) 2 by a newly introduced method of creative microscoping.…”

Some supercongruences arising from symbolic summation

“…By using the Dougall's formula, Long Recently, Guo and Schlosser [3, Theorem 2.3] established a partial q-analogue of (1.2) by using transformation formulas for basic hypergeometric series. One can refer to [1,2,4,5] for more q-analogues of congruences (q-congruence) for indefinite sums of binomial coefficients as well as hypergeometric series.…”