基本超几何函数

基本超几何函数是广义超几何函数的q模拟。

第一类基本超几何函数

${\displaystyle \;_{j}\phi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=0}^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k},q;q)_{n}}}\left((-1)^{n}q^{n \choose 2}\right)^{1+k-j}z^{n}}$ 

其中

${\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}}$ 

其中

${\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1}).}$ 

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第二类基本超几何函数

${\displaystyle \;_{j}\psi _{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]=\sum _{n=-\infty }^{\infty }{\frac {(a_{1},a_{2},\ldots ,a_{j};q)_{n}}{(b_{1},b_{2},\ldots ,b_{k};q)_{n}}}\left((-1)^{n}q^{n \choose 2}\right)^{k-j}z^{n}.}$ 

关系式

下列基本超几何函数在q->1时，化为超几何函数^[1]

${\displaystyle \lim _{q\to 1}\;_{j}\phi _{k}\left[{\begin{matrix}q^{a_{1}}&q^{a_{2}}&\ldots &q^{a_{j}}\\q^{b_{1}}&q^{b_{2}}&\ldots &q^{b_{k}}\end{matrix}};q,(q-1)*z\right]}$ = ${\displaystyle \;_{j}F_{k}\left[{\begin{matrix}a_{1}&a_{2}&\ldots &a_{j}\\b_{1}&b_{2}&\ldots &b_{k}\end{matrix}};q,z\right]}$ 

q二项式定理

下列公式是二项式定理的q模拟：

${\displaystyle _{1}\Phi _{0}([a],[];q;z)=}$ ${\displaystyle \sum _{n=0}^{\infty }}$ ${\displaystyle {\frac {(a;q)_{n}}{(q;q)_{n}}}}$ 

参考文献

1. Roelof KoeKoek, Peter Lesky,Rene Swarttouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues p15 Springer

• Fine, Nathan J., Basic hypergeometric series and applications, Mathematical Surveys and Monographs 27, Providence, R.I.: American Mathematical Society, 1988 [2015-01-25], ISBN 978-0-8218-1524-3, MR 0956465, （原始内容于2015-01-28） 
• Gasper, George; Rahman, Mizan, Basic hypergeometric series, Encyclopedia of Mathematics and its Applications 96 2nd, Cambridge University Press, 2004, ISBN 978-0-521-83357-8, MR 2128719, doi:10.2277/0521833574 
• Heine, Eduard, Über die Reihe ${\displaystyle 1+{\frac {(q^{\alpha }-1)(q^{\beta }-1)}{(q-1)(q^{\gamma }-1)}}x+{\frac {(q^{\alpha }-1)(q^{\alpha +1}-1)(q^{\beta }-1)(q^{\beta +1}-1)}{(q-1)(q^{2}-1)(q^{\gamma }-1)(q^{\gamma +1}-1)}}x^{2}+\cdots }$ , Journal für die reine und angewandte Mathematik, 1846, 32: 210–212 
• Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97–125.
• Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin