P-adic gamma function


In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by, though pointed out that implicitly used the same function. defined a p-adic analog Gp of log Γ. 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 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, can be extended uniquely to the whole. Here is the ring of p-adic integers. It comes by the definition that the values of are invertible in. This is so, because these values are products of integers not divisible by p, and this property holds after the continuous extension to. Thus. Here is the set of invertible p-adic integers.

Basic properties of \Gamma_p

The classical gamma function satisfies the functional equation for any. This has an analogue with respect to the Morita gamma function:
The Euler's reflection formula has its following simple counterpart in the p-adic case:
where is the first digit in the p-adic expansion of x, unless, in which case rather than 0.

Special values

and, in general,
At the Morita gamma function is related to the Legendre symbol:
It can also be seen, that hence as.
Other interesting special values come from the Gross–Koblitz formula, which was first proved by cohomological tools, and later was proved using more elementary methods. For example,
where denotes the root with first digit 3, and with we denote the root with first digit 2.
Another example is
where is the square root of in congruent to 1 modulo 3.

''p''-adic Raabe formula

The Raabe-formula for the classical Gamma function says that
This has an analogue for the Iwasawa logarithm of the Morita gamma function:
The ceiling function to be understood as the p-adic limit such that through rational integers.

Mahler expansion

The Mahler expansion is similarly important for p-adic functions as the Taylor expansion in classical analysis. The Mahler expansion of the p-adic gamma function is the following:
where the sequence is defined by the following identity: