Some q-congruences related to 3-adic valuations

Abstract

In 1992 Strauss, Shallit and Zagier proved that for any positive integer $a$ we have $$\sum _{k=0}^{3^a-1}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)=0(mod{3}^{2a})$$ and furthermore $$3^{-2a}}\sum_{k=0}^{3^a-1}\binom{2k}k=1 (mod 3).$$ Recently a $q$-analogue of the former congruence was conjectured by Guo and Zeng. In this paper we prove the Guo-Zeng conjecture and also give a $q$-analogue of the latter congruence.

9 pages

References in corpus (2)

• Various congruences involving binomial coefficients and higher-order Catalan numbers
• q-Analogs of some congruences involving Catalan numbers