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 of
is the ring of p-adic integers.
It follows from the definition that the values of
; this is because these values are products of integers not divisible by p, and this property holds after the continuous extension to
is the set of invertible p-adic integers.
The classical gamma function satisfies the functional equation
This has an analogue with respect to the Morita gamma function: The Euler's reflection formula
sin
has its following simple counterpart in the p-adic case: where
is the first digit in the p-adic expansion of x, unless
rather than 0. and, in general, At
the Morita gamma function is related to the Legendre symbol
( mod
[1]: 369 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.
denotes the square root with first digit 3, and
denotes the square root with first digit 2.
(Such specifications must always be done if we talk about roots.)
is the square root of
congruent to 1 modulo 3.
[3] The Raabe-formula for the classical Gamma function says that This has an analogue for the Iwasawa logarithm of the Morita gamma function:[4] The ceiling function to be understood as the p-adic limit
lim
through rational integers.
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:[1]: 374 where the sequence
is defined by the following identity: