基本超几何函数是广义超几何函数的q模拟。
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/274d2ef79218e289c26f62120cb6dcfcc6636248)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec36d7348a3c42b2d6475e1a8e7891406eba03b1)
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}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cbe9cbd694b153ad31c6388cb6ce546643a4989)
.
![{\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}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/501e00fbffa51da9bab6469656fbf853df7d794c)
下列基本超几何函数在q->1时,化为超几何函数[1]
= ![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54c9c5acefcc7bda7e1728534029ae8a3b83f26a)
下列公式是二项式定理的q模拟:
![{\displaystyle _{1}\Phi _{0}([a],[];q;z)=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a661b1eb79a4546dd7847df8a3a4b622d7c0456)
![{\displaystyle \sum _{n=0}^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c66edbb4a11f2bd77b3b2c469a68a77567811b10)
![{\displaystyle {\frac {(a;q)_{n}}{(q;q)_{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f669b24ee7a74cd76434075e26e364bccfc1be2a)
- ^ 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, (原始内容存档于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
, 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