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