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

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 ∈ Z + of Pascalʼs triangle is captured by the so-called Gaussian formula, which states that ∑ m = 0 n ( − 1 ) m ( n m ) q is 0 if n is odd, and is equal to ∏ k odd ( 1 − q k ) 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 ( mod p ) ( − 1 ) m ( n m ) ≡ 0 ( mod p ⌊ n − 1 p − 1 ⌋ ) .