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

TL;DR

This paper establishes prime modulus congruences for partial sums of generating series involving binomial coefficients rom specific algebraic substitutions, using closed-form series expressions.

Contribution

It introduces a method to derive congruences for partial sums of binomial generating series directly from their closed forms, covering shifted coefficients and specific algebraic substitutions.

Findings

Derived congruences modulo prime p for sums involving rom 0 to q and 0 to q/3.

Extended results to shifted binomial coefficients rom 3k+e.

Provided explicit formulas for sums with algebraic substitutions.

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.