"초기하급수(Hypergeometric series)"의 두 판 사이의 차이

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<math>\frac{\beta_{n+1}}{\beta_n} = \frac{1}{n+1}</math>
 
<math>\frac{\beta_{n+1}}{\beta_n} = \frac{1}{n+1}</math>
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계수의 비가 다음과 같이 주어지는 경우
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The ratio between consecutive coefficients now has the form <math>\frac{c(a_1+n)\dots(a_p+n)}{d(b_1+n)\dots(b_q+n)(1+n)}</math> where ''c'' and ''d'' are the leading coefficients of ''A'' and ''B''. The series then has the form :<math>1 + \frac{a_1\dots a_p}{b_1\dots b_q.1}\frac{cz}{d} + \frac{a_1\dots a_p}{b_1\dots b_q.1} \frac{(a_1+1)\dots(a_p+1)}{(b_1+1)\dots (b_q+1).2}\left(\frac{cz}{d}\right)^2+\dots</math>, or, by scaling z by the appropriate factor and rearranging, :<math>1 + \frac{a_1\dots a_p}{b_1\dots b_q}\frac{z}{1!} + \frac{a_1(a_1+1)\dots a_p(a_p+1)}{b_1(b_1+1)\dots b_q(b_q+1)}\frac{z^2}{2!}+\dots</math>. This has the form of an [[generating function|exponential generating function]]. The standard notation for this series is :<math>\,_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)</math>, although variations are sometimes used<ref>{{MathWorld|title=Generalized Hypergeometric Function|urlname=GeneralizedHypergeometricFunction}}</ref>. Using the rising factorial or [[Pochhammer symbol]]: :<math>(a)_n=a(a+1)(a+2)...(a+n-1),\,(a)_0 = 1</math>, this can be written :<math>\,_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\dots(a_p)_n}{(b_1)_n\dots(b_q)_n} \, \frac {z^n} {n!}</math>
  
 
 
 
 
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<h5>가우스의 초기하함수</h5>
 
<h5>가우스의 초기하함수</h5>
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<math>\,_2F_1(a,b;c;z)</math>
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<math>z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0</math>
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** George E. Andrews
 
** George E. Andrews
 
** This paper surveys recent applications of basic hypergeometric functions to partitions, number theory, finite vector spaces, combinatorial identities and physics.
 
** This paper surveys recent applications of basic hypergeometric functions to partitions, number theory, finite vector spaces, combinatorial identities and physics.
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* http://ko.wikipedia.org/wiki/
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* [http://ko.wikipedia.org/wiki/%EC%B4%88%EA%B8%B0%ED%95%98 http://ko.wikipedia.org/wiki/초기하]
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* http://en.wikipedia.org/wiki/hypergeometric_func
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* http://en.wikipedia.org/wiki/hypergeometric_functions
 
* http://www.wolframalpha.com/input/?i=
 
* http://www.wolframalpha.com/input/?i=
 
* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=

2009년 7월 1일 (수) 16:48 판

초기하급수
  • 두 연속한 계수의 비가 \(n\) 에 관한 유리함수인 멱급수를 초기하급수라 함. 즉,
    \(\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_n \beta_n z^n\)

\(\frac{\beta_{n+1}}{\beta_n} = \frac{A(n)}{B(n)}\) 이고 \({A(n)},{B(n)}\)은 n에 관한 다항식인 경우

  • 간단한 예

\(1+\frac{z}{1!}+\frac{z^2}{2!}+\frac{z^3}{3!}+\dots\)

\(\beta_n = \frac{1}{n!}\)

\(\frac{\beta_{n+1}}{\beta_n} = \frac{1}{n+1}\)

 

계수의 비가 다음과 같이 주어지는 경우

 

The ratio between consecutive coefficients now has the form \(\frac{c(a_1+n)\dots(a_p+n)}{d(b_1+n)\dots(b_q+n)(1+n)}\) where c and d are the leading coefficients of A and B. The series then has the form \[1 + \frac{a_1\dots a_p}{b_1\dots b_q.1}\frac{cz}{d} + \frac{a_1\dots a_p}{b_1\dots b_q.1} \frac{(a_1+1)\dots(a_p+1)}{(b_1+1)\dots (b_q+1).2}\left(\frac{cz}{d}\right)^2+\dots\], or, by scaling z by the appropriate factor and rearranging, \[1 + \frac{a_1\dots a_p}{b_1\dots b_q}\frac{z}{1!} + \frac{a_1(a_1+1)\dots a_p(a_p+1)}{b_1(b_1+1)\dots b_q(b_q+1)}\frac{z^2}{2!}+\dots\]. This has the form of an exponential generating function. The standard notation for this series is \[\,_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)\], although variations are sometimes used[1]. Using the rising factorial or Pochhammer symbol: \[(a)_n=a(a+1)(a+2)...(a+n-1),\,(a)_0 = 1\], this can be written \[\,_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\dots(a_p)_n}{(b_1)_n\dots(b_q)_n} \, \frac {z^n} {n!}\]

 

 

가우스의 초기하함수

\(\,_2F_1(a,b;c;z)\)

\(z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0\)

 

 

 

 

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  1. ↑ 틀:MathWorld
원본 주소 "https://wiki.mathnt.net/index.php?title=초기하급수(Hypergeometric_series)&oldid=19007"