数学において、q超幾何級数（qちょうきかきゅうすう、英: q-hypergeometric series, basic hypergeometric series）は、超幾何級数のq類似である。q超幾何級数は r ϕ s [ a 1 , a 2 , … , a r b 1 , b 2 , … , b s ; q , z ] = ∑ n = 0 ∞ ( a 1 , a 2 , … , a r ; q ) n ( b 1 , b 2 , … , b s , q ; q ) n ( ( − 1 ) n q ( n 2 ) ) s + 1 − r z n r ψ s [ a 1 , a 2 , … , a r b 1 , b 2 , … , b s ; q , z ] = ∑ n = − ∞ ∞ ( a 1 , a 2 , … , a r ; q ) n ( b 1 , b 2 , … , b s ; q ) n ( ( − 1 ) n q ( n 2 ) ) s − r z n {\displaystyle {\begin{aligned}{}_{r}\phi _{s}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{s}\end{matrix}};q,z\right]&=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{s},q;q)_{n}}}\left((-1)^{n}q^{\binom {n}{2}}\right)^{s+1-r}z^{n}\\{}_{r}\psi _{s}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{s}\end{matrix}};q,z\right]&=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{s};q)_{n}}}\left((-1)^{n}q^{\binom {n}{2}}\right)^{s-r}z^{n}\\\end{aligned}}} の形式で表される級数である[1]。中でも r ϕ r − 1 [ a 1 , a 2 , … , a r b 1 , b 2 , … , b r − 1 ; q , z ] = ∑ n = 0 ∞ ( a 1 , a 2 , … , a r ; q ) n ( b 1 , b 2 , … , b r − 1 , q ; q ) n z n r ψ r [ a 1 , a 2 , ⋯ , a r b 1 , b 2 , ⋯ , b r ; q , z ] = ∑ n = − ∞ ∞ ( a 1 , a 2 , … , a r ; q ) n ( b 1 , b 2 , … , b r ; q ) n z n {\displaystyle {\begin{aligned}{}_{r}\phi _{r-1}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{r-1}\end{matrix}};q,z\right]&=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{r-1},q;q)_{n}}}z^{n}\\{}_{r}\psi _{r}\left[{\begin{matrix}a_{1},a_{2},\dotsb ,a_{r}\\b_{1},b_{2},\dotsb ,b_{r}\end{matrix}};q,z\right]&=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{r};q)_{n}}}z^{n}\\\end{aligned}}} が多く研究されている。ただし、 ( a 1 , a 2 , … , a r ; q ) n = ( a 1 ; q ) n ( a 2 ; q ) n ⋯ ( a r ; q ) n {\displaystyle (a_{1},a_{2},\dotsc ,a_{r};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\dotsm (a_{r};q)_{n}} であり、ここで ( a ; q ) n = { ∏ 0 ≤ k < n ( 1 − a q k ) ( n > 0 ) 1 ( n = 0 ) ∏ n ≤ k < 0 ( 1 − a q k ) − 1 ( n < 0 ) {\displaystyle (a;q)_{n}={\begin{cases}\prod _{0\leq k<n}(1-aq^{k})&(n>0)\\1&(n=0)\\\prod _{n\leq k<0}(1-aq^{k})^{-1}&(n<0)\end{cases}}} はqポッホハマー記号である。なお、厳密にいうと、右辺の級数がq超幾何級数であり、左辺の記号は級数の和によって定義されるq超幾何関数を表すものである。 Remove ads [1]Wolfram Mathworld: q-Hypergeometric Function 堀田良之・渡辺敬一・庄司俊明・三町勝久: 群論の進化, 代数学百科, I, 朝倉書店, 2004 年, ISBN 4-254-11099-5 Gasper, G., Rahman, M. (2004). Basic hypergeometric series. en:Cambridge university press. Andrews, G. E., Askey, R., & Roy, R. (1999). Special functions. en:Cambridge university press. q二項定理 ハイネの和公式 ラマヌジャンの和公式 qザールシュッツの和公式Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for EdgeWikiwand for FirefoxRemove ads
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![{\displaystyle {\begin{aligned}{}_{r}\phi _{r-1}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{r-1}\end{matrix}};q,z\right]&=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{r-1},q;q)_{n}}}z^{n}\\{}_{r}\psi _{r}\left[{\begin{matrix}a_{1},a_{2},\dotsb ,a_{r}\\b_{1},b_{2},\dotsb ,b_{r}\end{matrix}};q,z\right]&=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{r};q)_{n}}}z^{n}\\\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/a3d2fc385b1227fc7456d5fcd96b5a5bfa46ba53)
![{\displaystyle {\begin{aligned}{}_{r}\phi _{s}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{s}\end{matrix}};q,z\right]&=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{s},q;q)_{n}}}\left((-1)^{n}q^{\binom {n}{2}}\right)^{s+1-r}z^{n}\\{}_{r}\psi _{s}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{s}\end{matrix}};q,z\right]&=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{s};q)_{n}}}\left((-1)^{n}q^{\binom {n}{2}}\right)^{s-r}z^{n}\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1f5f0ed59af4340dfa6667da28cb0d84195d8db3)
![{\displaystyle {\begin{aligned}{}_{r}\phi _{r-1}\left[{\begin{matrix}a_{1},a_{2},\dotsc ,a_{r}\\b_{1},b_{2},\dotsc ,b_{r-1}\end{matrix}};q,z\right]&=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{r-1},q;q)_{n}}}z^{n}\\{}_{r}\psi _{r}\left[{\begin{matrix}a_{1},a_{2},\dotsb ,a_{r}\\b_{1},b_{2},\dotsb ,b_{r}\end{matrix}};q,z\right]&=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\dotsc ,a_{r};q)_{n}}{(b_{1},b_{2},\dotsc ,b_{r};q)_{n}}}z^{n}\\\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/a3d2fc385b1227fc7456d5fcd96b5a5bfa46ba53)
