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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.12535 |
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| _version_ | 1866910193492688896 |
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| author | Buckingham, Paul |
| author_facet | Buckingham, Paul |
| contents | Let $r$ be a non-zero rational number. In a paper in the Transactions of the AMS in 2023, O'Desky and Richman gave a construction of a $p$-adic incomplete gamma-function $Γ_p(\cdot,r)$ for each prime $p$ for which $|r - 1|_p < 1$. Aside from the special case where $r = 1$, only finitely many primes satisfy that condition for a given $r$, so it is desirable to lessen this restriction. In the present paper, we give a construction that works under the much weaker condition that $|r|_p = 1$ using a $p$-adic integral transform we introduced in our paper of 2024 in Acta Arithmetica, which we interpret here as a $p$-adic analogue of an incomplete Mellin transform. For any given $r$, the condition $|r|_p = 1$ holds for all \emph{except} finitely many primes $p$.
Our approach emphasizes the parallels between the complex and $p$-adic constructions, explaining how a $p$-adic integration-by-parts formula takes the place of complex integration by parts in the proof of the recurrence relations for the $p$-adic incomplete gamma-functions. We introduce a two-variable $p$-adic transform for the task, extending our earlier $p$-adic integral transform. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12535 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $p$-adic incomplete Mellin transforms and $p$-adic incomplete gamma-functions Buckingham, Paul Number Theory 11S80 Let $r$ be a non-zero rational number. In a paper in the Transactions of the AMS in 2023, O'Desky and Richman gave a construction of a $p$-adic incomplete gamma-function $Γ_p(\cdot,r)$ for each prime $p$ for which $|r - 1|_p < 1$. Aside from the special case where $r = 1$, only finitely many primes satisfy that condition for a given $r$, so it is desirable to lessen this restriction. In the present paper, we give a construction that works under the much weaker condition that $|r|_p = 1$ using a $p$-adic integral transform we introduced in our paper of 2024 in Acta Arithmetica, which we interpret here as a $p$-adic analogue of an incomplete Mellin transform. For any given $r$, the condition $|r|_p = 1$ holds for all \emph{except} finitely many primes $p$. Our approach emphasizes the parallels between the complex and $p$-adic constructions, explaining how a $p$-adic integration-by-parts formula takes the place of complex integration by parts in the proof of the recurrence relations for the $p$-adic incomplete gamma-functions. We introduce a two-variable $p$-adic transform for the task, extending our earlier $p$-adic integral transform. |
| title | On $p$-adic incomplete Mellin transforms and $p$-adic incomplete gamma-functions |
| topic | Number Theory 11S80 |
| url | https://arxiv.org/abs/2512.12535 |