P진 감마함수(p-adic gamma function)

정의

• 자연수 \(n\) 에 대하여 다음과 같이 정의

\[\Gamma_p(n)=(-1)^n\prod_{(i,p)=1}^{n-1} i\]

• 이를 \(\mathbb{Z}_p\)로 연속함수로 확장하여, p-adic 감마함수를 얻음

예

• 아래는 $p=2,3,5,7$일 때, $\Gamma_p$의 값이다

$$ \begin{array}{c|cccc} n & \Gamma _2(n) & \Gamma _3(n) & \Gamma _5(n) & \Gamma _7(n) \\ \hline -10 & -\frac{1}{945} & -\frac{1}{22400} & \frac{1}{72576} & -\frac{1}{518400} \\ -9 & \frac{1}{945} & -\frac{1}{2240} & -\frac{1}{72576} & -\frac{1}{51840} \\ -8 & \frac{1}{105} & \frac{1}{2240} & -\frac{1}{8064} & -\frac{1}{5760} \\ -7 & -\frac{1}{105} & \frac{1}{280} & -\frac{1}{1008} & -\frac{1}{720} \\ -6 & -\frac{1}{15} & \frac{1}{40} & -\frac{1}{144} & \frac{1}{720} \\ -5 & \frac{1}{15} & -\frac{1}{40} & -\frac{1}{24} & \frac{1}{120} \\ -4 & \frac{1}{3} & -\frac{1}{8} & \frac{1}{24} & \frac{1}{24} \\ -3 & -\frac{1}{3} & -\frac{1}{2} & \frac{1}{6} & \frac{1}{6} \\ -2 & -1 & \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ -1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 1 & -1 & -1 & -1 & -1 \\ 2 & 1 & 1 & 1 & 1 \\ 3 & -1 & -2 & -2 & -2 \\ 4 & 3 & 2 & 6 & 6 \\ 5 & -3 & -8 & -24 & -24 \\ 6 & 15 & 40 & 24 & 120 \\ 7 & -15 & -40 & -144 & -720 \\ 8 & 105 & 280 & 1008 & 720 \\ 9 & -105 & -2240 & -8064 & -5760 \\ 10 & 945 & 2240 & 72576 & 51840 \end{array} $$

기본적인 성질

• \(x\in p\mathbb{Z}_p\) 일 때, \(\Gamma_p(x+1)=-\Gamma_p(x)\)
• \(x\not \in p\mathbb{Z}_p\) 일 때, \(\Gamma_p(x+1)=-x\Gamma_p(x)\)
• \(x \equiv y \pmod {p^r}\) 이면 \(\Gamma_p(x)\equiv \Gamma_p(y) \pmod {p^r}\)
• \(p>3\) 이면 \(|\Gamma_p(x)-\Gamma_p(y)|_p \leq |x-y|_p\)

테이블

• 소수 $p$와 정수 $x$에 대하여, $\operatorname{ord}_p x$를 $a\equiv 0\pmod {p^m}$을 만족하는 최대의 $m\in \mathbb{Z}_{\geq 0}$으로 정의하자
• 유리수 $x=a/b$에 대해서는 $\operatorname{ord}_p x:=\operatorname{ord}_p a-\operatorname{ord}_p b$

$$ \begin{array}{c|c|c} \{x,y\} & \operatorname{ord}_5 (x-y) & \operatorname{ord}_5 \left(\Gamma _5(x)-\Gamma _5(y)\right) \\ \hline \{-5,-4\} & 0 & 0 \\ \{-5,-3\} & 0 & 1 \\ \{-5,-2\} & 0 & 0 \\ \{-5,-1\} & 0 & 2 \\ \{-5,0\} & 1 & 2 \\ \{-5,1\} & 0 & 0 \\ \{-5,2\} & 0 & 2 \\ \{-5,3\} & 0 & 0 \\ \{-5,4\} & 0 & 1 \\ \{-5,5\} & 1 & 2 \\ \{-4,-3\} & 0 & 0 \\ \{-4,-2\} & 0 & 0 \\ \{-4,-1\} & 0 & 0 \\ \{-4,0\} & 0 & 0 \\ \{-4,1\} & 1 & 2 \\ \{-4,2\} & 0 & 0 \\ \{-4,3\} & 0 & 0 \\ \{-4,4\} & 0 & 0 \\ \{-4,5\} & 0 & 0 \\ \{-3,-2\} & 0 & 0 \\ \{-3,-1\} & 0 & 1 \\ \{-3,0\} & 0 & 1 \\ \{-3,1\} & 0 & 0 \\ \{-3,2\} & 1 & 1 \\ \{-3,3\} & 0 & 0 \\ \{-3,4\} & 0 & 1 \\ \{-3,5\} & 0 & 1 \\ \{-2,-1\} & 0 & 0 \\ \{-2,0\} & 0 & 0 \\ \{-2,1\} & 0 & 0 \\ \{-2,2\} & 0 & 0 \\ \{-2,3\} & 1 & 1 \\ \{-2,4\} & 0 & 0 \\ \{-2,5\} & 0 & 0 \\ \{-1,0\} & 0 & \infty \\ \{-1,1\} & 0 & 0 \\ \{-1,2\} & 0 & \infty \\ \{-1,3\} & 0 & 0 \\ \{-1,4\} & 1 & 1 \\ \{-1,5\} & 0 & 2 \\ \{0,1\} & 0 & 0 \\ \{0,2\} & 0 & \infty \\ \{0,3\} & 0 & 0 \\ \{0,4\} & 0 & 1 \\ \{0,5\} & 1 & 2 \\ \{1,2\} & 0 & 0 \\ \{1,3\} & 0 & 0 \\ \{1,4\} & 0 & 0 \\ \{1,5\} & 0 & 0 \\ \{2,3\} & 0 & 0 \\ \{2,4\} & 0 & 1 \\ \{2,5\} & 0 & 2 \\ \{3,4\} & 0 & 0 \\ \{3,5\} & 0 & 0 \\ \{4,5\} & 0 & 1 \end{array} $$

반사공식

• \(p\neq 2\)이고, \(x\in \mathbb{Z}_p\) 에 대하여 다음 반사공식이 성립

\[\Gamma_p(x)\Gamma_p(1-x)=(-1)^{l(x)}\] 여기서 \(x\equiv l(x) \pmod p\), \(l(x)\in \{1,2,\cdots, p\}\) $$ \begin{array}{c|cccc} x & \Gamma _3(x) & \Gamma _3(1-x) & \Gamma _3(1-x) \Gamma _3(x) & (-1)^{l(x)} \\ \hline -10 & -\frac{1}{22400} & -22400 & 1 & 1 \\ -9 & -\frac{1}{2240} & 2240 & -1 & -1 \\ -8 & \frac{1}{2240} & -2240 & -1 & -1 \\ -7 & \frac{1}{280} & 280 & 1 & 1 \\ -6 & \frac{1}{40} & -40 & -1 & -1 \\ -5 & -\frac{1}{40} & 40 & -1 & -1 \\ -4 & -\frac{1}{8} & -8 & 1 & 1 \\ -3 & -\frac{1}{2} & 2 & -1 & -1 \\ -2 & \frac{1}{2} & -2 & -1 & -1 \\ -1 & 1 & 1 & 1 & 1 \\ 0 & 1 & -1 & -1 & -1 \\ 1 & -1 & 1 & -1 & -1 \\ 2 & 1 & 1 & 1 & 1 \\ 3 & -2 & \frac{1}{2} & -1 & -1 \\ 4 & 2 & -\frac{1}{2} & -1 & -1 \\ 5 & -8 & -\frac{1}{8} & 1 & 1 \\ 6 & 40 & -\frac{1}{40} & -1 & -1 \\ 7 & -40 & \frac{1}{40} & -1 & -1 \\ 8 & 280 & \frac{1}{280} & 1 & 1 \\ 9 & -2240 & \frac{1}{2240} & -1 & -1 \\ 10 & 2240 & -\frac{1}{2240} & -1 & -1 \end{array} $$

역사

• 수학사 연표

관련된 항목들

• P진 해석학(p-adic analysis)
• 감마함수

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxeS1SUlM0MWxYNWc/edit

관련논문

• The Gross Koblitz formula revisited
  • Robert, Alain M. , Rend. Sem. Mat. Univ. Padova 105 (2001) 157 170.
• p-adic gamma functions and their applications
  • Jack Diamond, 1984
• The p-adic gamma function and congruences of Atkin and. Swinnerton-Dyer
  • L. van Hamme, Groupe d'étude d'analyse ultramétrique, 9e année 81/82, Fasc. 3 no. J17-6p
• Gauss Sums and the p-adic Γ-function
  • Benedict H. Gross and Neal Koblitz, The Annals of Mathematics, Second Series, Vol. 109, No. 3 (May, 1979), pp. 569-581
• The $p$-Adic Log Gamma Function and $p$-Adic Euler Constants
  • Jack Diamond, Transactions of the American Mathematical Society, Vol. 233, (Oct., 1977), pp. 321-337
• A p-adic analogue of the $\Gamma$-function
  • Morita, Yasuo, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics, Vol.22(1975), No.2, Page 255-266