A generalization of the Gaussian formula and a q-analog of Fleck's congruence

TL;DR

This paper extends the Gaussian formula to a q-analog and introduces a new q-binomial congruence that unifies classical identities like Fleck's congruence, enriching the combinatorial and number-theoretic framework.

Contribution

It presents a novel q-binomial congruence that generalizes the Gaussian formula and incorporates Fleck's congruence, bridging classical and polynomial identities.

Findings

Established a q-analog of the Gaussian formula

Derived a new q-binomial congruence unifying classical results

Enhanced understanding of q-binomial coefficient identities

Abstract

The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of the alternating sum across row n of Pascal's triangle is captured by the so-called Gaussian Formula. In this paper, we find a q-binomial congruence which synthesizes this result and Fleck's congruence for binomial coefficients.