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 , 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 ${\mathbb{Z}}_p$) 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 $x\in ℂ\setminus {\mathbb{Z}}_{\le 0}$. This has an analogue with respect to the Morita gamma function:

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

The Euler's reflection formula $\Gamma (x)\Gamma (1-x)=\frac{\pi }{\sin (\pi x)}$ 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,

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

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

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

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

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

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

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

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

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

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

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

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

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 $x_n\to x$ 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:Template:Rp

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

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

• Boyarsky, Maurizio (1980), "p-adic gamma functions and Dwork cohomology", Transactions of the American Mathematical Society, 257 (2): 359–369, doi:10.2307/1998301, ISSN 0002-9947, JSTOR 1998301, MR 0552263
• Diamond, Jack (1977), "The p-adic log gamma function and p-adic Euler constants", Transactions of the American Mathematical Society, 233: 321–337, doi:10.2307/1997840, ISSN 0002-9947, JSTOR 1997840, MR 0498503
• Diamond, Jack (1984), "p-adic gamma functions and their applications", in Chudnovsky, David V.; Chudnovsky, Gregory V.; Cohn, Henry; et al. (eds.), Number theory (New York, 1982), Lecture Notes in Math., vol. 1052, Berlin, New York: Springer-Verlag, pp. 168–175, doi:10.1007/BFb0071542, ISBN 978-3-540-12909-7, MR 0750664
• Dwork, Bernard (1964), "On the zeta function of a hypersurface. II", Annals of Mathematics, Second Series, 80 (2): 227–299, doi:10.2307/1970392, ISSN 0003-486X, JSTOR 1970392, MR 0188215
• Morita, Yasuo (1975), "A p-adic analogue of the Γ-function", Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics, 22 (2): 255–266, hdl:2261/6494, ISSN 0040-8980, MR 0424762
• Overholtzer, Gordon (1952), "Sum functions in elementary p-adic analysis", American Journal of Mathematics, 74 (2): 332–346, doi:10.2307/2371998, ISSN 0002-9327, JSTOR 2371998, MR 0048493