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October 16, 2025Open Access

Congruences for sums involving rkk

Key Points

Congruences are derived modulo p^2 for sums including binomial coefficients, marking a key finding.
The research explores sums $inom{rk}{k}x^k/k$ for ranges $0<k<p$ and $0<k<p/r$, demonstrating their importance.
Two distinct approaches are utilized, offering complementary strengths for deriving the results on congruences.
The findings express congruences in terms of finite polylogarithms related to algebraic numbers, indicating future exploration potential.

Abstract

We primarily investigate congruences modulo p for finite sums of the form ₖrkkxᵏ/k over the ranges 0<k<p and 0<k<p/r, where p is a prime larger than the positive integer r. Here x is an indeterminate, thus allowing specialization to numerical congruences where x takes certain algebraic numbers as values. We employ two different approaches that have complementary strengths. In particular, we obtain congruences modulo p² for the sum ₀<₊<₏rkkxᵏ, expressed in terms of finite polylogarithms of certain quantities related to x.

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Cite This Study

Mattarei et al. (Wed,) studied this question.

synapsesocial.com/papers/68f147cc724575985c3fd1dc https://doi.org/https://doi.org/10.48550/arxiv.2505.09849

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