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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.05728 |
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| _version_ | 1866915279106211840 |
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| author | Wang, Rong-Hua Zhong, Michael X. X. |
| author_facet | Wang, Rong-Hua Zhong, Michael X. X. |
| contents | In this paper, we apply the power-partible reduction to study arithmetic properties of sums involving Delannoy numbers $D_k$ and polynomials $D_k(z)$. Let $v\in\bN$ and $p$ be an odd prime. It is proved that, for any $z\in\bZ\setminus\{0,-1\}$, there exist $c_v\in z^{-v}\bZ[z]$ and $\tilde{c}_v\in (z+1)^{-v}\bZ[z]$, both free of $p$ and can be determined mechanically, such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2v}D_k(z)\equiv c_v \left(\frac{-z}{p}\right) \pmod {p} \end{equation*} if $\gcd(p,z)=1$ and \begin{equation*} \sum_{k=0}^{p-1}(-1)^k(2k+1)^{2v}D_k(z)\equiv \tilde{c}_v \left(\frac{z+1}{p}\right) \pmod {p} \end{equation*} if $\gcd(p,z+1)=1$. Here $(-)$ denotes the Legendre symbol.
When $n$ is a power of $2$, we find there exist odd integers $ρ_v$ and even integers $\tildeρ_v$, both independent of $n$ and can be determined mechanically, such that \[ \sum_{k=0}^{n-1}(2k+1)^{2v+1}D_k\equiv ρ_v n \pmod {n^3} \] and \[ \sum_{k=0}^{n-1}(-1)^k(2k+1)^{2v+1}D_k\equiv \tildeρ_v n^2 \pmod {n^3}. \] The case $v=1$ in the last congruence confirms a conjecture of Guo and Zeng in 2012. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_05728 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Congruences for sums of Delannoy numbers and polynomials Wang, Rong-Hua Zhong, Michael X. X. Combinatorics In this paper, we apply the power-partible reduction to study arithmetic properties of sums involving Delannoy numbers $D_k$ and polynomials $D_k(z)$. Let $v\in\bN$ and $p$ be an odd prime. It is proved that, for any $z\in\bZ\setminus\{0,-1\}$, there exist $c_v\in z^{-v}\bZ[z]$ and $\tilde{c}_v\in (z+1)^{-v}\bZ[z]$, both free of $p$ and can be determined mechanically, such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2v}D_k(z)\equiv c_v \left(\frac{-z}{p}\right) \pmod {p} \end{equation*} if $\gcd(p,z)=1$ and \begin{equation*} \sum_{k=0}^{p-1}(-1)^k(2k+1)^{2v}D_k(z)\equiv \tilde{c}_v \left(\frac{z+1}{p}\right) \pmod {p} \end{equation*} if $\gcd(p,z+1)=1$. Here $(-)$ denotes the Legendre symbol. When $n$ is a power of $2$, we find there exist odd integers $ρ_v$ and even integers $\tildeρ_v$, both independent of $n$ and can be determined mechanically, such that \[ \sum_{k=0}^{n-1}(2k+1)^{2v+1}D_k\equiv ρ_v n \pmod {n^3} \] and \[ \sum_{k=0}^{n-1}(-1)^k(2k+1)^{2v+1}D_k\equiv \tildeρ_v n^2 \pmod {n^3}. \] The case $v=1$ in the last congruence confirms a conjecture of Guo and Zeng in 2012. |
| title | Congruences for sums of Delannoy numbers and polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2505.05728 |