Some q-congruences related to 3-adic valuations

TL;DR

This paper proves a conjecture by Guo and Zeng regarding q-analogues of certain 3-adic valuations related to binomial sums, extending classical congruences with new q-analogue results.

Contribution

The paper confirms the Guo-Zeng conjecture and introduces a q-analogue of a known 3-adic valuation congruence involving binomial coefficients.

Findings

Proof of the Guo-Zeng q-conjecture.

Introduction of a q-analogue for a classical 3-adic valuation congruence.

Extension of known binomial sum congruences to q-analogues.

Abstract

In 1992 Strauss, Shallit and Zagier proved that for any positive integer $a$ we have $$\sum_{k=0}^{3^a-1}\binom{2k}{k}=0 (mod 3^{2a})$$ 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.