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arXiv · 2210.06146

Congruences for partial sums of the generating series for $\binom{3k}{k}$

Sandro Mattarei·Roberto Tauraso
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Abstract

We produce congruences modulo a prime $p>3$ for sums $\sum_k\binom{3k}{k}x^k$ over ranges $0\le k<q$ and $0\le k<q/3$, where $q$ is a power of $p$. Here $x$ equals either $c^2/(1-c)^3$, or $4s^2/\bigl(27(s^2-1)\bigr)$, where $c$ and $s$ are indeterminates. In the former case we deal more generally with shifted binomial coefficients $\binom{3k+e}{k}$. Our method derives such congruences directly from closed forms for the corresponding series.

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Sandro Mattarei, Roberto Tauraso. 2022-10-12. Congruences for partial sums of the generating series for $\binom{3k}{k}$. https://arxiv.org/abs/2210.06146

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