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, though pointed out that  implicitly used the same function. defined a p-adic analog Gp of log&thinsp;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 $$\mathbb{Z}_p$$) such that


 * $$\Gamma_p(x) = (-1)^x \prod_{0<i<x,\ p \,\nmid\, i} i$$

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 $$\mathbb{Z}_p$$, $$\Gamma_p(x)$$ 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 $$\Gamma_p(\mathbb{Z})$$ 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 $$\Gamma_p:\mathbb{Z}_p\to\mathbb{Z}_p^\times$$. 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 $$\Gamma(x+1) = x\Gamma(x)$$ for any $$x\in\mathbb{C}\setminus\mathbb{Z}_{\le0}$$. This has an analogue with respect to the Morita gamma function:


 * $$\frac{\Gamma_p(x+1)}{\Gamma_p(x)}=\begin{cases} -x, & \mbox{if } x \in \mathbb{Z}_p^\times \\ -1,  & \mbox{if } x\in p\mathbb{Z}_p. \end{cases}$$

The Euler's reflection formula $$\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin{(\pi x)}}$$ 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 $$x \in p\mathbb{Z}_p$$, 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,
 * $$\Gamma_p(n+1)=\frac{(-1)^{n+1}n!}{[n/p]!p^{[n/p]}}\quad(n\ge2).$$

At $$x=\frac12$$ the Morita gamma function is related to the Legendre symbol $$\left(\frac{a}{p}\right)$$:
 * $$\Gamma_p\left(\frac12\right)^2 = -\left(\frac{-1}{p}\right).$$

It can also be seen, that $$\Gamma_p(p^n)\equiv1\pmod{p^n},$$ hence $$\Gamma_p(p^n)\to1$$ as $$n\to\infty$$.

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,
 * $$\Gamma_5\left(\frac14\right)^2=-2+\sqrt{-1},$$
 * $$\Gamma_7\left(\frac13\right)^3=\frac{1-3\sqrt{-3}}{2},$$

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
 * $$\Gamma_3\left(\frac18\right)\Gamma_3\left(\frac38\right)=-(1+\sqrt{-2}),$$

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


 * $$\int_0^1\log\Gamma(x+t)dt=\frac12\log(2\pi)+x\log x-x.$$

This has an analogue for the Iwasawa logarithm of the Morita gamma function:
 * $$\int_{\mathbb{Z}_p}\log\Gamma_p(x+t)dt=(x-1)(\log\Gamma_p)'(x)-x+\left\lceil\frac{x}{p}\right\rceil\quad(x\in\mathbb{Z}_p).$$

The ceiling function to be understood as the p-adic limit $$\lim_{n\to\infty}\left\lceil\frac{x_n}{p}\right\rceil$$ 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:


 * $$\Gamma_p(x+1)=\sum_{k=0}^\infty a_k\binom{x}{k},$$

where the sequence $$a_k$$ is defined by the following identity:
 * $$\sum_{k=0}^\infty(-1)^{k+1}a_k\frac{x^k}{k!}=\frac{1-x^p}{1-x}\exp\left(x+\frac{x^p}{p}\right).$$