The sum ${\sum }_{k=0}^{q-1}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)$ for q a power of 3

Abstract

We prove that ${\sum }_{k=0}^{q-1}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)\equiv q^2(\mathrm{mod}3q^2)$ if q>1 is a power of 3, as recently conjectured by Z.W. Sun and R. Tauraso. Our more precise result actually implies that the value of $(1/q^2){\sum }_{k=0}^{q-1}\left(\genfrac{}{}{0ex}{}{2k}{k}\right)$ modulo a fixed arbitrary power of 3 is independent of q, for q a power of 3 large enough, and shows how such value can be efficiently computed.

arXiv comment10 pages