![{\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}}](//wikimedia.org/api/rest_v1/media/math/render/svg/274d2ef79218e289c26f62120cb6dcfcc6636248)
其中
其中
.
![{\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}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/501e00fbffa51da9bab6469656fbf853df7d794c)
下列基本超幾何函數在q->1時，化為超幾何函數[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]}](//wikimedia.org/api/rest_v1/media/math/render/svg/cc0a5e591d342d2bc13051fe13a097aeafd4599d)
= ![{\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]}](//wikimedia.org/api/rest_v1/media/math/render/svg/54c9c5acefcc7bda7e1728534029ae8a3b83f26a)
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, （原始內容存檔於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
![{\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}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/274d2ef79218e289c26f62120cb6dcfcc6636248)
![{\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}.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/501e00fbffa51da9bab6469656fbf853df7d794c)
![{\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]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/cc0a5e591d342d2bc13051fe13a097aeafd4599d)
![{\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]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/54c9c5acefcc7bda7e1728534029ae8a3b83f26a)
![{\displaystyle _{1}\Phi _{0}([a],[];q;z)=}](http://wikimedia.org/api/rest_v1/media/math/render/svg/2a661b1eb79a4546dd7847df8a3a4b622d7c0456)
