{{DISPLAYTITLE:''p''-adic gamma function}} In mathematics, the '''''p''-adic gamma function''' Γ<sub>''p''</sub> is a function of a ''p''-adic variable analogous to the gamma function. It was first explicitly defined by {{harvtxt|Morita|1975}}, though {{harvtxt|Boyarsky|1980}} pointed out that {{harvtxt|Dwork|1964}} implicitly used the same function. {{harvtxt|Diamond|1977}} defined a ''p''-adic analog ''G''<sub>''p''</sub> of log Γ. {{harvtxt|Overholtzer|1952}} 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 <math>\mathbb{Z}_p</math>) such that
:<math>\Gamma_p(x) = (-1)^x \prod_{0<i<x,\ p \,\nmid\, i} i</math>
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 <math>\mathbb{Z}_p</math>, <math>\Gamma_p(x)</math> can be extended uniquely to the whole of <math>\mathbb{Z}_p</math>. Here <math>\mathbb{Z}_p</math> is the ring of ''p''-adic integers. It follows from the definition that the values of <math>\Gamma_p(\mathbb{Z})</math> are invertible in <math>\mathbb{Z}_p</math>; this is because these values are products of integers not divisible by ''p'', and this property holds after the continuous extension to <math>\mathbb{Z}_p</math>. Thus <math>\Gamma_p:\mathbb{Z}_p\to\mathbb{Z}_p^\times</math>. Here <math>\mathbb{Z}_p^\times</math> is the set of invertible ''p''-adic integers.
==Basic properties of the ''p''-adic gamma function==
The classical gamma function satisfies the functional equation <math>\Gamma(x+1) = x\Gamma(x)</math> for any <math>x\in\mathbb{C}\setminus\mathbb{Z}_{\le0}</math>. This has an analogue with respect to the Morita gamma function:
:<math>\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}</math>
The Euler's reflection formula <math>\Gamma(x)\Gamma(1-x) = \frac{\pi}{\sin{(\pi x)}}</math> has its following simple counterpart in the ''p''-adic case: :<math>\Gamma_p(x)\Gamma_p(1-x) = (-1)^{x_0},</math> where <math>x_0</math> is the first digit in the ''p''-adic expansion of ''x'', unless <math>x \in p\mathbb{Z}_p</math>, in which case <math>x_0 = p</math> rather than 0.
==Special values==
:<math>\Gamma_p(0)=1,</math> :<math>\Gamma_p(1)=-1,</math> :<math>\Gamma_p(2)=1,</math> :<math>\Gamma_p(3)=-2,</math> and, in general, :<math>\Gamma_p(n+1)=\frac{(-1)^{n+1}n!}{[n/p]!p^{[n/p]}}\quad(n\ge2).</math>
At <math>x=\frac12</math> the Morita gamma function is related to the Legendre symbol <math>\left(\frac{a}{p}\right)</math>: :<math>\Gamma_p\left(\frac12\right)^2 = -\left(\frac{-1}{p}\right).</math>
It can also be seen, that <math>\Gamma_p(p^n)\equiv1\pmod{p^n},</math> hence <math>\Gamma_p(p^n)\to1</math> as <math>n\to\infty</math>.<ref name=RobertBook/>{{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.<ref name='Robert'>{{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 }}</ref> For example, :<math>\Gamma_5\left(\frac14\right)^2=-2+\sqrt{-1},</math> :<math>\Gamma_7\left(\frac13\right)^3=\frac{1-3\sqrt{-3}}{2},</math> where <math>\sqrt{-1}\in\mathbb{Z}_5</math> denotes the square root with first digit 3, and <math>\sqrt{-3}\in\mathbb{Z}_7</math> denotes the square root with first digit 2. (Such specifications must always be done if we talk about roots.)
Another example is :<math>\Gamma_3\left(\frac18\right)\Gamma_3\left(\frac38\right)=-(1+\sqrt{-2}),</math> where <math>\sqrt{-2}</math> is the square root of <math>-2</math> in <math>\mathbb{Q}_3</math> congruent to 1 modulo 3.<ref>{{cite book|first = H. | last = Cohen | title=Number Theory | volume = 2| publisher = Springer Science+Business Media | location = New York | date=2007 | page = 406}}</ref>
==''p''-adic Raabe formula==
The Raabe formula for the classical Gamma function says that
:<math>\int_0^1\log\Gamma(x+t)dt=\frac12\log(2\pi)+x\log x-x.</math>
This has an analogue for the Iwasawa logarithm of the Morita gamma function:<ref>{{cite journal | last1 = Cohen | first1 = Henri | author-link1=Henri Cohen (number theorist) | 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 }}</ref> :<math>\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).</math> The ceiling function to be understood as the ''p''-adic limit <math>\lim_{n\to\infty}\left\lceil\frac{x_n}{p}\right\rceil</math> such that <math>x_n\to x</math> 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:<ref name=RobertBook/>{{rp|374}}
:<math>\Gamma_p(x+1)=\sum_{k=0}^\infty a_k\binom{x}{k},</math> where the sequence <math>a_k</math> is defined by the following identity: :<math>\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).</math>
==See also==
==References== *{{Citation | last1=Boyarsky | first1=Maurizio | author-link1=Bernard Dwork | title=p-adic gamma functions and Dwork cohomology | doi=10.2307/1998301 |mr=552263 | year=1980 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=257 | issue=2 | pages=359–369| jstor=1998301 }} *{{Citation | last1=Diamond | first1=Jack | title=The p-adic log gamma function and p-adic Euler constants | jstor=1997840 |mr=0498503 | year=1977 | journal=Transactions of the American Mathematical Society | issn=0002-9947 | volume=233 | pages=321–337 | doi=10.2307/1997840}} *{{Citation | last1=Diamond | first1=Jack | editor1-last=Chudnovsky | editor1-first=David V. | editor1-link=Chudnovsky brothers | editor2-last=Chudnovsky | editor2-first=Gregory V. | editor3-last=Cohn | editor3-first=Henry |display-editors = 3 | editor4-last=Nathanson | editor4-first=Melvyn B. | title=Number theory (New York, 1982) | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Math. | isbn=978-3-540-12909-7 | doi=10.1007/BFb0071542 |mr=750664 | year=1984 | volume=1052 | chapter=p-adic gamma functions and their applications | pages=168–175}} *{{Citation | last1=Dwork | first1=Bernard | author-link1=Bernard Dwork | title=On the zeta function of a hypersurface. II | jstor=1970392 |mr=0188215 | year=1964 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=80 | issue=2 | pages=227–299 | doi=10.2307/1970392}} *{{Citation | last1=Morita | first1=Yasuo | title=A p-adic analogue of the Γ-function |mr=0424762 | year=1975 | journal=Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics | issn=0040-8980 | volume=22 | issue=2 | pages=255–266| hdl=2261/6494 }} *{{Citation | last1=Overholtzer | first1=Gordon | title=Sum functions in elementary p-adic analysis | jstor=2371998 |mr=0048493 | year=1952 | journal=American Journal of Mathematics | issn=0002-9327 | volume=74 | issue=2 | pages=332–346 | doi=10.2307/2371998}}
<references>
<ref name=RobertBook> {{cite book | first = Alain M. | last = Robert | title = A course in p-adic analysis | publisher = Springer-Verlag | location = New York | date=2000}} </ref>
</references>
Category:Number theory Category:P-adic numbers