Congruences for sums involving $\binom{rk}{k}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911520555794432 |
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| author | Mattarei, Sandro Tauraso, Roberto |
| author_facet | Mattarei, Sandro Tauraso, Roberto |
| contents | We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ 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 $p^2$ for the sum $\sum_{0<k<p}\binom{rk}{k}x^k$, expressed in terms of finite polylogarithms of certain quantities related to $x$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_09849 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Congruences for sums involving $\binom{rk}{k}$ Mattarei, Sandro Tauraso, Roberto Number Theory 11A07 (Primary) 05A10, 05A15 (Secondary) We primarily investigate congruences modulo $p$ for finite sums of the form $\sum_k\binom{rk}{k}x^k/k$ 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 $p^2$ for the sum $\sum_{0<k<p}\binom{rk}{k}x^k$, expressed in terms of finite polylogarithms of certain quantities related to $x$. |
| title | Congruences for sums involving $\binom{rk}{k}$ |
| topic | Number Theory 11A07 (Primary) 05A10, 05A15 (Secondary) |
| url | https://arxiv.org/abs/2505.09849 |