p-adic gamma function
In mathematics, the p-adic gamma function Γp(s) is a function of a p-adic variable s analogous to the gamma function. It was first explicitly defined by Morita (1975), though Boyarsky (1980) pointed out that Dwork (1964) implicitly used the same function. Diamond (1977) defined a p-adic analog Gp(s) of log Γ(s). Overholtzer (1952) had previously given a definition of a different p-adic analogue of the gamma function, but his function does not have satisfactory properties and is not used much.
Definition
The p-adic gamma function is the unique continuous function of a p-adic integer s such that
for positive integers s, where the product is restricted to integers i not divisible by p.
See also
References
- Boyarsky, Maurizio (1980), "p-adic gamma functions and Dwork cohomology", Transactions of the American Mathematical Society 257 (2): 359–369, doi:10.2307/1998301, ISSN 0002-9947, JSTOR 1998301, MR 552263
- Diamond, Jack (1977), "The p-adic log gamma function and p-adic Euler constants", Transactions of the American Mathematical Society 233: 321–337, doi:10.2307/1997840, ISSN 0002-9947, JSTOR 1997840, MR 0498503
- Diamond, Jack (1984), "p-adic gamma functions and their applications", in Chudnovsky, David V.; Chudnovsky, Gregory V.; Cohn, Henry; et al., Number theory (New York, 1982), Lecture Notes in Math. 1052, Berlin, New York: Springer-Verlag, pp. 168–175, doi:10.1007/BFb0071542, ISBN 978-3-540-12909-7, MR 750664
- Dwork, Bernard (1964), "On the zeta function of a hypersurface. II", Annals of Mathematics. Second Series 80: 227–299, doi:10.2307/1970392, ISSN 0003-486X, JSTOR 1970392, MR 0188215
- Morita, Yasuo (1975), "A p-adic analogue of the Γ-function", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics 22 (2): 255–266, ISSN 0040-8980, MR 0424762
- Overholtzer, Gordon (1952), "Sum functions in elementary p-adic analysis", American Journal of Mathematics 74: 332–346, doi:10.2307/2371998, ISSN 0002-9327, JSTOR 2371998, MR 0048493
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