Mixing
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。
目录
1 第一类基本超几何函数
2 第二类基本超几何函数
3 关系式
4 q二项式定理
5 参考文献
第一类基本超几何函数
- jϕk[a1a2…ajb1b2…bk;q,z]=∑n=0∞(a1,a2,…,aj;q)n(b1,b2,…,bk,q;q)n((−1)nq(n2))1+k−jzn{displaystyle ;_{j}phi _{k}left[{begin{matrix}a_{1}&a_{2}&ldots &a_{j}\b_{1}&b_{2}&ldots &b_{k}end{matrix}};q,zright]=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}}
![{displaystyle ;_{j}phi _{k}left[{begin{matrix}a_{1}&a_{2}&ldots &a_{j}\b_{1}&b_{2}&ldots &b_{k}end{matrix}};q,zright]=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
- (a1,a2,…,am;q)n=(a1;q)n(a2;q)n…(am;q)n{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
- (a;q)n=∏k=0n−1(1−aqk)=(1−a)(1−aq)(1−aq2)⋯(1−aqn−1).{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
- jψk[a1a2…ajb1b2…bk;q,z]=∑n=−∞∞(a1,a2,…,aj;q)n(b1,b2,…,bk;q)n((−1)nq(n2))k−jzn.{displaystyle ;_{j}psi _{k}left[{begin{matrix}a_{1}&a_{2}&ldots &a_{j}\b_{1}&b_{2}&ldots &b_{k}end{matrix}};q,zright]=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}.}
![{displaystyle ;_{j}psi _{k}left[{begin{matrix}a_{1}&a_{2}&ldots &a_{j}\b_{1}&b_{2}&ldots &b_{k}end{matrix}};q,zright]=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)
关系式
下列基本超几何函数在q->1时，化为超几何函数[1]
limq→1jϕk[qa1qa2…qajqb1qb2…qbk;q,(q−1)∗z]{displaystyle lim _{qto 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)*zright]}= jFk[a1a2…ajb1b2…bk;q,z]{displaystyle ;_{j}F_{k}left[{begin{matrix}a_{1}&a_{2}&ldots &a_{j}\b_{1}&b_{2}&ldots &b_{k}end{matrix}};q,zright]}
![{displaystyle ;_{j}F_{k}left[{begin{matrix}a_{1}&a_{2}&ldots &a_{j}\b_{1}&b_{2}&ldots &b_{k}end{matrix}};q,zright]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54c9c5acefcc7bda7e1728534029ae8a3b83f26a)
q二项式定理
下列公式是二项式定理的q模拟：
1Φ0([a],[];q;z)={displaystyle _{1}Phi _{0}([a],;q;z)=}∑n=0∞{displaystyle sum _{n=0}^{infty }}
(a;q)n(q;q)n{displaystyle {frac {(a;q)_{n}}{(q;q)_{n}}}}
参考文献
^ 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, ISBN 978-0-8218-1524-3, MR 956465
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 1+(qα−1)(qβ−1)(q−1)(qγ−1)x+(qα−1)(qα+1−1)(qβ−1)(qβ+1−1)(q−1)(q2−1)(qγ−1)(qγ+1−1)x2+⋯{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 参数
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Eduard Heine, Theorie der Kugelfunctionen, (1878) 1, pp 97–125.- Eduard Heine, Handbuch die Kugelfunctionen. Theorie und Anwendung (1898) Springer, Berlin
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