p-adic gamma function

The p-adic gamma function is the unique continuous function of a p-adic integer x (with values in ${\displaystyle \mathbb {Z} _{p}}$) such that

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

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

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

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

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

${\displaystyle \Gamma _{p}(x)\Gamma _{p}(1-x)=(-1)^{x_{0}},}$

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

${\displaystyle \Gamma _{p}(0)=1,}$

${\displaystyle \Gamma _{p}(1)=-1,}$

${\displaystyle \Gamma _{p}(2)=1,}$

${\displaystyle \Gamma _{p}(3)=-2,}$

and, in general,

${\displaystyle \Gamma _{p}(n+1)={\frac {(-1)^{n+1}n!}{[n/p]!p^{[n/p]}}}\quad (n\geq 2).}$

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

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

It can also be seen, that ${\displaystyle \Gamma _{p}(p^{n})\equiv 1{\pmod {p^{n}}},}$ hence ${\displaystyle \Gamma _{p}(p^{n})\to 1}$ as ${\displaystyle n\to \infty }$.[1]^: 369

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.[2] For example,

${\displaystyle \Gamma _{5}\left({\frac {1}{4}}\right)^{2}=-2+{\sqrt {-1}},}$

${\displaystyle \Gamma _{7}\left({\frac {1}{3}}\right)^{3}={\frac {1-3{\sqrt {-3}}}{2}},}$

where ${\displaystyle {\sqrt {-1}}\in \mathbb {Z} _{5}}$ denotes the square root with first digit 3, and ${\displaystyle {\sqrt {-3}}\in \mathbb {Z} _{7}}$ denotes the square root with first digit 2. (Such specifications must always be done if we talk about roots.)

Another example is

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

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

The Raabe-formula for the classical Gamma function says that

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

This has an analogue for the Iwasawa logarithm of the Morita gamma function:[4]

${\displaystyle \int _{\mathbb {Z} _{p}}\log \Gamma _{p}(x+t)dt=(x-1)(\log \Gamma _{p})'(x)-x+\left\lceil {\frac {x}{p}}\right\rceil \quad (x\in \mathbb {Z} _{p}).}$

The ceiling function to be understood as the p-adic limit ${\displaystyle \lim _{n\to \infty }\left\lceil {\frac {x_{n}}{p}}\right\rceil }$ such that ${\displaystyle x_{n}\to x}$ through rational integers.

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:[1]^: 374

${\displaystyle \Gamma _{p}(x+1)=\sum _{k=0}^{\infty }a_{k}{\binom {x}{k}},}$

where the sequence ${\displaystyle a_{k}}$ is defined by the following identity:

${\displaystyle \sum _{k=0}^{\infty }(-1)^{k+1}a_{k}{\frac {x^{k}}{k!}}={\frac {1-x^{p}}{1-x}}\exp \left(x+{\frac {x^{p}}{p}}\right).}$

• Gross–Koblitz formula

• 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