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 $\mathbb_p$) such that

:$\Gamma_p(x) = (-1)^x \prod_ i$

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 $\mathbb_p$, $\Gamma_p(x)$ can be extended uniquely to the whole of $\mathbb_p$. Here $\mathbb_p$ is the ring of ''p''-adic integers. It follows from the definition that the values of $\Gamma_p(\mathbb)$ are invertible in $\mathbb_p$; this is because these values are products of integers not divisible by ''p'', and this property holds after the continuous extension to $\mathbb_p$. Thus $\Gamma_p:\mathbb_p\to\mathbb_p^\times$. Here $\mathbb_p^\times$ 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 $x\in\mathbb\setminus\mathbb_$. This has an analogue with respect to the Morita gamma function:

:$\frac=\begin -x, & \mbox x \in \mathbb_p^\times \\ -1, & \mbox x\in p\mathbb_p. \end$

The Euler's reflection formula $\Gamma(x)\Gamma(1-x) = \frac$ has its following simple counterpart in the ''p''-adic case:
:$\Gamma_p(x)\Gamma_p(1-x) = (-1)^,$
where $x_0$ is the first digit in the ''p''-adic expansion of ''x'', unless $x \in p\mathbb_p$, 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,
:$\Gamma_p(n+1)=\frac\quad(n\ge2).$

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

It can also be seen, that $\Gamma_p(p^n)\equiv1\pmod,$ 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,
:$\Gamma_5\left(\frac14\right)^2=-2+\sqrt,$
:$\Gamma_7\left(\frac13\right)^3=\frac,$
where $\sqrt\in\mathbb_5$ denotes the square root with first digit 3, and $\sqrt\in\mathbb_7$ denotes the square root with first digit 2. (Such specifications must always be done if we talk about roots.)

Another example is
:$\Gamma_3\left(\frac18\right)\Gamma_3\left(\frac38\right)=-(1+\sqrt),$
where $\sqrt$ is the square root of $-2$ in $\mathbb_3$ 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:
:$\int_\log\Gamma_p(x+t)dt=(x-1)(\log\Gamma_p)'(x)-x+\left\lceil\frac\right\rceil\quad(x\in\mathbb_p).$
The ceiling function to be understood as the ''p''-adic limit $\lim_\left\lceil\frac\right\rceil$ 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:

:$\Gamma_p(x+1)=\sum_^\infty a_k\binom,$
where the sequence $a_k$ is defined by the following identity:
:$\sum_^\infty(-1)^a_k\frac=\frac\exp\left(x+\frac\right).$

See also

* Gross–Koblitz formula

References

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{{reflist, refs=

{{cite book , first = Alain M. , last = Robert , title = A course in p-adic analysis , publisher = Springer-Verlag , location = New York , date=2000

Number theory
P-adic numbers