A $q$-microscope for supercongruences

TL;DR

This paper introduces a novel $q$-microscope approach to analyze hypergeometric sums at roots of unity, leading to new supercongruences for truncated sums, bridging $q$-series and number theory.

Contribution

It develops a $q$-analogue framework to derive supercongruences for hypergeometric sums, connecting asymptotic $q$-behavior with classical number-theoretic congruences.

Findings

Derived supercongruences for truncated hypergeometric sums.

Established polynomial congruences from $q$-hypergeometric asymptotics.

Connected $q$-series analysis with classical 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 $$ \sum_{n=0}^\infty\frac{\binom{4n}{2n}{\binom{2n}{n}}^2}{2^{8n}3^{2n}}\,(8n+1) =\frac{2\sqrt{3}}{\pi}, $$ of the two supercongruences $$ S(p-1)\equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3} \quad\text{and}\quad S\Bigl(\frac{p-1}2\Bigr) \equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3}, $$ valid for all primes $p>3$, where $S(N)$ denotes the truncation of the infinite sum at the $N$-th place and $(\frac{-3}{\cdot})$ stands for the quadratic character modulo $3$.