p-adic gamma function

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

:

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

The classical gamma function satisfies the functional equation $\backslash Gamma(x+1)\; =\; x\backslash Gamma(x)$ for any $x\backslash in\backslash mathbb\{C\}\backslash setminus\backslash mathbb\{Z\}\_\{\backslash le0\}$. This has an analogue with respect to the Morita gamma function:

:

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

:$\backslash Gamma\_p(x)\backslash Gamma\_p(1-x)\; =\; (-1)^\{x\_0\},$

where $x\_0$ is the first digit in the p-adic expansion of x, unless $x\; \backslash in\; p\backslash mathbb\{Z\}\_p$, in which case $x\_0\; =\; p$ rather than 0.

:$\backslash Gamma\_p(0)=1,$

:$\backslash Gamma\_p(1)=-1,$

:$\backslash Gamma\_p(2)=1,$

:$\backslash Gamma\_p(3)=-2,$

and, in general,

:$\backslash Gamma\_p(n+1)=\backslash frac\{(-1)^\{n+1\}n!\}\{[n/p]!p^\{[n/p]\}\}\backslash quad(n\backslash ge2).$

At $x=\backslash frac12$ the Morita gamma function is related to the Legendre symbol $\backslash left(\backslash frac\{a\}\{p\}\backslash right)$:

:$\backslash Gamma\_p\backslash left(\backslash frac12\backslash right)^2\; =\; -\backslash left(\backslash frac\{-1\}\{p\}\backslash right).$

It can also be seen, that $\backslash Gamma\_p(p^n)\backslash equiv1\backslash pmod\{p^n\},$ hence $\backslash Gamma\_p(p^n)\backslash to1$ as $n\backslash to\backslash infty$.{{rp|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.{{cite journal | last1 = Robert | first1 = Alain M. | title = The Gross-Koblitz formula revisited | url=http://www.numdam.org/item?id=RSMUP_2001__105__157_0 | mr=1834987 | journal = Rendiconti del Seminario Matematico della Università di Padova. The Mathematical Journal of the University of Padova | issn=0041-8994 | year = 2001 | volume = 105 | pages = 157–170 | doi=10.1016/j.jnt.2009.08.005| hdl = 2437/90539 | hdl-access = free }} For example,

:$\backslash Gamma\_5\backslash left(\backslash frac14\backslash right)^2=-2+\backslash sqrt\{-1\},$

:$\backslash Gamma\_7\backslash left(\backslash frac13\backslash right)^3=\backslash frac\{1-3\backslash sqrt\{-3\}\}\{2\},$

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

Another example is

:$\backslash Gamma\_3\backslash left(\backslash frac18\backslash right)\backslash Gamma\_3\backslash left(\backslash frac38\backslash right)=-(1+\backslash sqrt\{-2\}),$

where $\backslash sqrt\{-2\}$ is the square root of $-2$ in $\backslash mathbb\{Q\}\_3$ congruent to 1 modulo 3.{{cite book|first = H. | last = Cohen | title=Number Theory | volume = 2| publisher = Springer Science+Business Media | location = New York | date=2007 | page = 406}}

The Raabe-formula for the classical Gamma function says that

:$\backslash int\_0^1\backslash log\backslash Gamma(x+t)dt=\backslash frac12\backslash log(2\backslash pi)+x\backslash log\; x-x.$

This has an analogue for the Iwasawa logarithm of the Morita gamma function:{{cite journal | last1 = Cohen | first1 = Henri | last2 = Eduardo | first2 = Friedman | title = Raabe's formula for p-adic gamma and zeta functions | mr = 2401225 | journal = Annales de l'Institut Fourier | year = 2008 | volume = 88 |issue = 1 | pages = 363–376 | doi=10.5802/aif.2353 | url = http://www.numdam.org/item/AIF_2008__58_1_363_0/ | hdl = 10533/139530 | hdl-access = free }}

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

The ceiling function to be understood as the p-adic limit $\backslash lim\_\{n\backslash to\backslash infty\}\backslash left\backslash lceil\backslash frac\{x\_n\}\{p\}\backslash right\backslash rceil$ such that $x\_n\backslash to\; x$ 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:{{rp|374}}

:$\backslash Gamma\_p(x+1)=\backslash sum\_\{k=0\}^\backslash infty\; a\_k\backslash binom\{x\}\{k\},$

where the sequence $a\_k$ is defined by the following identity:

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

• Gross–Koblitz formula

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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}}

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Category:Number theory

Category:P-adic numbers