Some q-congruences related to 3-adic valuations

Details Some q-congruences related to 3-adic valuations Some q-congruences related to 3-adic valuations

2009-10-22

arxiv.org/abs/0910.4170v2

Mathematics

Number Theory

9 pages

Scientific paper

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.

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