p-adic gamma function

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 Template:Harvtxt, though Template:Harvtxt pointed out that Template:Harvtxt implicitly used the same function. Template:Harvtxt defined a p-adic analog G_p of log Γ. Template:Harvtxt 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 ${\mathbb{Z}}_p$) such that

${\mathrm{\Gamma }}_p(x)=(-1{)}^x\prod _{0<i<x,p\nmid i}i$

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

Basic properties of the p-adic gamma function

The classical gamma function satisfies the functional equation $\mathrm{\Gamma }(x+1)=x\mathrm{\Gamma }(x)$ for any $x\in ℂ\setminus {\mathbb{Z}}_{\le 0}$. This has an analogue with respect to the Morita gamma function:

$\frac{{\mathrm{\Gamma }}_p(x+1)}{{\mathrm{\Gamma }}_p(x)}=\{\begin{array}{cc}-x,& \text{if }x\in {\mathbb{Z}}_p^{\times }\\ -1,& \text{if }x\in p{\mathbb{Z}}_p.\end{array}$

The Euler's reflection formula $\mathrm{\Gamma }(x)\mathrm{\Gamma }(1-x)=\frac{\pi }{\sin (\pi x)}$ has its following simple counterpart in the p-adic case:

${\mathrm{\Gamma }}_p(x){\mathrm{\Gamma }}_p(1-x)=(-1{)}^{x_0},$

where $x_0$ is the first digit in the p-adic expansion of x, unless $x\in p{\mathbb{Z}}_p$, in which case $x_0=p$ rather than 0.

Special values

${\mathrm{\Gamma }}_p(0)=1,$

${\mathrm{\Gamma }}_p(1)=-1,$

${\mathrm{\Gamma }}_p(2)=1,$

${\mathrm{\Gamma }}_p(3)=-2,$

and, in general,

${\mathrm{\Gamma }}_p(n+1)=\frac{(-1{)}^{n+1}n!}{[n/p]!p^{[n/p]}}(n\ge 2).$

At $x=\frac{1}{2}$ the Morita gamma function is related to the Legendre symbol $\left(\frac{a}{p}\right)$:

${\mathrm{\Gamma }}_p{\left(\frac{1}{2}\right)}^2=-\left(\frac{-1}{p}\right).$

It can also be seen, that ${\mathrm{\Gamma }}_p(p^n)\equiv 1(\mathrm{mod}{p}^n),$ hence ${\mathrm{\Gamma }}_p(p^n)\to 1$ as $n\to \mathrm{\infty }$.Template:Rp

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,

${\mathrm{\Gamma }}_5{\left(\frac{1}{4}\right)}^2=-2+\sqrt{-1},$

${\mathrm{\Gamma }}_7{\left(\frac{1}{3}\right)}^3=\frac{1-3\sqrt{-3}}{2},$

where $\sqrt{-1}\in {\mathbb{Z}}_5$ denotes the square root with first digit 3, and $\sqrt{-3}\in {\mathbb{Z}}_7$ denotes the square root with first digit 2. (Such specifications must always be done if we talk about roots.)

Another example is

${\mathrm{\Gamma }}_3\left(\frac{1}{8}\right){\mathrm{\Gamma }}_3\left(\frac{3}{8}\right)=-(1+\sqrt{-2}),$

where $\sqrt{-2}$ is the square root of $-2$ in ${\mathbb{Q}}_3$ congruent to 1 modulo 3.

p-adic Raabe formula

The Raabe formula for the classical Gamma function says that

${\int }_0^1\log \mathrm{\Gamma }(x+t)dt=\frac{1}{2}\log (2\pi )+x\log x-x.$

This has an analogue for the Iwasawa logarithm of the Morita gamma function:

${\int }_{{\mathbb{Z}}_p}\log {\mathrm{\Gamma }}_p(x+t)dt=(x-1)(\log {\mathrm{\Gamma }}_p{)}^{\prime }(x)-x+\lceil \frac{x}{p}\rceil (x\in {\mathbb{Z}}_p).$

The ceiling function to be understood as the p-adic limit $\underset{n\to \mathrm{\infty }}{lim}\lceil \frac{x_n}{p}\rceil$ such that $x_n\to x$.

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:Template:Rp

${\mathrm{\Gamma }}_p(x+1)=\sum _{k=0}^{\mathrm{\infty }}{a}_k(\genfrac{}{}{0ex}{}{x}{k}),$

where the sequence $a_k$ is defined by the following identity:

$\sum _{k=0}^{\mathrm{\infty }}(-1{)}^{k+1}{a}_k\frac{x^k}{k!}=\frac{1-x^p}{1-x}\exp (x+\frac{x^p}{p}).$

See also

• Gross–Koblitz formula

References

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