A generalization of the Gaussian formula and a q-analog of Fleckʼs congruence

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∈Z+ of Pascalʼs triangle is captured by the so-called Gaussian formula, which states that ∑m=0n(−1)m(nm)q is 0 if n is odd, and is equal to ∏kodd(1−qk) if n is even. In this paper, we find a q-binomial congruence which synthesizes this result and Fleckʼs congruence for binomial coefficients, which asserts that for n,p∈Z+, with p a prime,∑m≡j(modp)(−1)m(nm)≡0(modp⌊n−1p−1⌋).