A q-microscope for supercongruences

By Victor J. W. Guo, Wadim Zudilin. Advances in Mathematics, 2018

258 citations recorded by the source corpus.

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 ∞ ( 4 n 2 n ) ( 2 n n ) 2 2 8 n 3 2 n ( 8 n + 1 ) = 2 3 π , of the two supercongruences S ( p − 1 ) ≡ p ( − 3 p ) ( mod p 3 ) and S ( p − 1 2 ) ≡ p ( − 3 p ) ( mod p 3 ) , 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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a2a484bc582a51ae79da9208ae94e1ed80dff4d4

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Citations in the corpus

On a congruence involving $q$-Catalan numbers (2019, methodology): “In the past few years, q-analogues of congruences (q-congruence) for indefinite sums of binomial coefficients as well as hypergeometric series attracted many experts’ attention (see, for example, [2–6,9,10]).”
Some congruences related to a congruence of Van Hamme (2019, methodology): “For more supercongruences and q-supercongruences, we refer the reader to [4, 6–20, 26, 29, 30].”
Congruences for q-Binomial Coefficients (2019, background): “croscope setup’ (when q-congruences for truncated hypergeometric sums are read oﬀ from the asymptotics of their non-terminating versions, usually equipped with extra parameters, at roots of unity—see [5]) for Straub’s q-congruence [8], [9, Theorem 2.2], an bn q ≡ a b qn2 − b(a −b) a b n2 − 1 2”
A family of q-hypergeometric congruences modulo the fourth power of a cyclotomic polynomial (2019, background): “During the past few years, many congruences and supercongruences were generalized to the q-setting by various authors (see, for instance, [5–18, 25, 26, 30, 32]).”
q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon (2018, result): “The sequence an(q) was studied by Krattenthaler, Rivoal and Zudilin [KRZ06] and Zheng [Zhe11] in the case f(x, y) = (x− y)2 and recently by Straub [Str18] for general f .”
On a q-deformation of modular forms (2018, methodology): “A principal objective of our earlier work [21] was demonstration that the hypergeometric evaluations like (1.”
Supercongruences for polynomial analogs of the Apéry numbers (2018, methodology): “Zudilin [22], who apply it with remarkable success to supercongruences in the context of Ramanujan-type formulae for 1/π.”