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 Morita (1975), though Boyarsky (1980) pointed out that Dwork (1964) implicitly used the same function. Diamond (1977) defined a p-adic analog Gp 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 (with values in ) 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 Edit
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.
and, in general,
At the Morita gamma function is related to the Legendre symbol:
It can also be seen, that hence as .:369
where denotes the root with first digit 3, and with we denote the root with first digit 2. (Such specifications must always be done if we talk about roots.)
Another example is
where is the square root of in congruent to 1 modulo 3.
p-adic Raabe formulaEdit
The Raabe-formula for the classical Gamma function says that
The ceiling function to be understood as the p-adic limit such that through rational integers.
where the sequence is defined by the following identity:
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