p-adic gamma function

In mathematics, the p-adic gamma function Γ_p(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 G_p(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

$\Gamma_p(s)=(-1)^s\prod_{0<i<s,\ p\nmid i}i$

for positive integers s, where the product is restricted to integers i not divisible by p.

See also

• Gross–Koblitz formula

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