P-adic gamma function

In mathematics, the p-adic gamma function Γ_p is a function of a p-adic variable 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 of log Γ. 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.

The p-adic gamma function is the unique continuous function of a p-adic integer x such that

for positive integers x, where the product is restricted to integers i not divisible by p. As the positive integers are dense with respect to the p-adic topology in ${\displaystyle \mathbb {Z} _{p}}$, ${\displaystyle \Gamma _{p}(x)}$ can be extended uniquely to the whole ${\displaystyle \mathbb {Z} _{p}}$. Here ${\displaystyle \mathbb {Z} _{p}}$ is the ring of p-adic integers. It comes by the definition that the values of ${\displaystyle \Gamma _{p}(\mathbb {Z} )}$ are invertible in ${\displaystyle \mathbb {Z} _{p}}$. This is so, because these values are products of integers not divisible by p, and this property holds after the continuous extension to ${\displaystyle \mathbb {Z} _{p}}$. Thus ${\displaystyle \Gamma _{p}:\mathbb {Z} _{p}\to \mathbb {Z} _{p}^{\times }}$. Here ${\displaystyle \mathbb {Z} _{p}^{\times }}$ is the set of invertible p-adic integers.

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