\(p\)-adic incomplete gamma functions and Artin-Hasse-type series. (English) Zbl 1528.33025
Summary: We define and study a \(p\)-adic analogue of the incomplete gamma function related to Morita’s \(p\)-adic gamma function. We also discuss a combinatorial identity related to the Artin-Hasse series, which is a special case of the exponential principle in combinatorics. From this we deduce a curious \(p\)-adic property of \(\#\mathrm{Hom}(G,S_n)\) for a topologically finitely generated group \(G\), using a characterization of \(p\)-adic continuity for certain functions \(f \colon \mathbb Z_{>0} \to \mathbb Q_p\) due to A. O’Desky and D. H. Richman [Sémin. Lothar. Comb. 86B, Article 81, 10 p. (2022; Zbl 1524.33090)]. In the end, we give an exposition of some standard properties of the Artin-Hasse series.
MSC:
| 33E50 | Special functions in characteristic \(p\) (gamma functions, etc.) |
| 33B20 | Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals) |
| 05A05 | Permutations, words, matrices |
| 11S80 | Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) |
Keywords:
\(p\)-adic analysis; \(p\)-adic gamma function; Artin-Hasse series; exponential principle; symmetric groupsCitations:
Zbl 1524.33090References:
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