P-adic gamma function explained

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, though pointed out that implicitly used the same function. defined a p-adic analog G_p 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 (with values in

Z _{p} ) such that\Gamma _{p(x)} = ( -1 ) ^{x} \prod _{0<i<x, p} i

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

Z _{p} , \Gamma _{p(x)} can be extended uniquely to the whole of Z _{p} . Here Z _{p} is the ring of p-adic integers. It follows from the definition that the values of \Gamma _{p(Z)} are invertible in Z _{p} ; this is because these values are products of integers not divisible by p, and this property holds after the continuous extension to Z _{p} . Thus \Gamma _{p:Z} _{p\toZ} ^{x} ^{p} . Here Z ^{x} _{p} is the set of invertible p-adic integers.

Basic properties of the p-adic gamma function

The classical gamma function satisfies the functional equation

\Gamma(x +1 ) = x\Gamma(x) for any x\in C \setminus Z _{\le0} . This has an analogue with respect to the Morita gamma function:(\Gamma _{p(x+1)})/(\Gamma _{p(x)}) = \begin{cases} - x, & if x \in Z ^{x} _{p} \ -1 , & if x\in p Z _{p.} \end{cases} \Gamma(x)\Gamma( 1- x) = (\pi)/(\sin{(\pi x)) } has its following simple counterpart in the p-adic case:\Gamma _{p(x)\Gamma} _{p(1-x)} = ( -1 ) ^{x _{0}} , where x _{0} is the first digit in the p-adic expansion of x, unless x \in p Z _{p} , in which case x _{0} = p rather than 0.

Special values

\Gamma _{p(0)=1,} \Gamma _{p(1)=-1,} \Gamma _{p(2)=1,} \Gamma _{p(3)=-2,} and, in general,\Gamma _{p(n +1 ) = (( -1 ) ^{n+1} n!)/([n/p]!p ^{[n/p]})} (n\ge 2 ).

At

x = (12)/() the Morita gamma function is related to the Legendre symbol \left( (a)/(p) \right) :\Gamma _{p\left( (12 \right))/(^{2})} =- \left( (-1)/(p) \right).

It can also be seen, that

\Gamma ^{n)\equiv 1 \pmod{p} _{p(p} ^{n},} hence \Gamma ^{n)\to 1} _{p(p} as n\to inf ty .

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.^[1] For example,

\Gamma _{5 \left( (14 \right))/(^{2=-2+\sqrt{-1},})} \Gamma _{7 \left( (13 \right))/(^{3= (1-3 \sqrt{ -3)/()})} },where \sqrt{ -1 }\in Z _{5} denotes the square root with first digit 3, and \sqrt{ -3 }\in Z _{7} denotes the square root with first digit 2. (Such specifications must always be done if we talk about roots.)

Another example is

\Gamma _{3 \left( (18 \right)\Gamma)/(^{3 \left( (38 \right) =- ( 1+ \sqrt{ -2 }),)/()})} where \sqrt{ -2 } is the square root of -2 in Q _{3} congruent to 1 modulo 3.^[2]

p-adic Raabe formula

The Raabe-formula for the classical Gamma function says that

\int ^{1log \Gamma(x + t)dt = (12log ( 2 \pi) + x log)/(x - x.)} _{0}

This has an analogue for the Iwasawa logarithm of the Morita gamma function:^[3]

\int _{Z _{p}} log \Gamma _{p(x+t)dt=(x-1)(log\Gamma} ^{p)'(x) - x + \left\lceil (x)/(p)} \right\rceil (x\in Z _{p).} The ceiling function to be understood as the p-adic limit \lim _{n\toinfty} \left\lceil (x _{n})/(p) \right\rceil such that x _{n\to} x 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:

\Gamma _{p(x+1)=\sum} ^{inf ty} ^{k =0} a _{k\binom{x}{k},} where the sequence a _{k} is defined by the following identity:\sum ^{inf ty( -1 )} _{k =0} ^{k+1} a _{k (x ^{k})/(k!)} = (1- x ^{p})/(\exp\left(x + 1- x) (x ^{p})/(p) \right).

See also

• Gross–Koblitz formula

Notes and References

1. Robert . Alain M. . The Gross-Koblitz formula revisited . 1834987 . Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova . 0041-8994 . 2001 . 105 . 157–170 . 10.1016/j.jnt.2009.08.005. 2437/90539 . free .
2. Book: Cohen, H. . Number Theory . 2. . New York . 2007 . 406.
3. Cohen . Henri . Eduardo . Friedman . Raabe's formula for p-adic gamma and zeta functions . 2401225 . Annales de l'Institut Fourier . 2008 . 88 . 1 . 363–376 . 10.5802/aif.2353 . 10533/139530 . free .