P-adic gamma function
Encyclopedia
In mathematics, the p-adic gamma function Γp(s) is a function of a p-adic variable s 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(s) of log Γ(s). 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.
for positive integers s, where the product is restricted to integers i not divisible by p.
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
. It was first explicitly defined by , though pointed out that implicitly used the same function. defined a p-adic analog Gp(s) of log Γ(s). 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 thatfor positive integers s, where the product is restricted to integers i not divisible by p.