P진 감마함수(p-adic gamma function)

수학노트
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개요

  • 감마함수와 유사한 성질을 가지는 p진 해석학의 함수


정의

  • 자연수 \(n\) 에 대하여 다음과 같이 $\Gamma_p$의 값을 정의

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

  • 이를 \(\mathbb{Z}_p\)에서의 연속함수로 확장하여, p-adic 감마함수 $\Gamma_p:\mathbb{Z}_p\to \mathbb{Z}_p^{\times}$를 얻음


  • 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


관련된 항목들


매스매티카 파일 및 계산 리소스


사전 형태의 자료


에세이, 리뷰, 강의노트

  • Diamond, Jack. 1984. “P-Adic Gamma Functions and Their Applications.” In Number Theory, edited by David V. Chudnovsky, Gregory V. Chudnovsky, Harvey Cohn, and Melvin B. Nathanson, 168–75. Lecture Notes in Mathematics 1052. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0071542.


관련논문

  • Barman, Rupam, and Neelam Saikia. “Supercongruences for Truncated Hypergeometric Series and P-Adic Gamma Function.” arXiv:1507.07391 [math], July 27, 2015. http://arxiv.org/abs/1507.07391.
  • Fuselier, Jenny G., and Dermot McCarthy. “Hypergeometric Type Identities in the $p$-Adic Setting and Modular Forms.” arXiv:1407.6670 [math], July 24, 2014. http://arxiv.org/abs/1407.6670.
  • Robert, Alain M., The Gross Koblitz formula revisited, Rend. Sem. Mat. Univ. Padova 105 (2001) 157 170.
  • 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
  • Gross, Benedict H., and Neal Koblitz. 1979. “Gauss Sums and the p-Adic $\Gamma$-Function.” The Annals of Mathematics 109 (3): 569. doi:10.2307/1971226.
  • Diamond, Jack. 1977. “The P-Adic Log Gamma Function and P-Adic Euler Constants.” Transactions of the American Mathematical Society 233 (October): 321. doi:10.2307/1997840.
  • Morita, Yasuo, A p-adic analogue of the $\Gamma$-function, Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics, Vol.22(1975), No.2, Page 255-266