查看“基本超几何函数”的源代码

'''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==
:<math>\;_{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)^nq^{n\choose 2}\right)^{1+k-j}z^n</math>

其中 

:<math>(a_1,a_2,\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \ldots (a_m;q)_n</math>
其中
:<math>(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1}).</math>

.

==第二类基本超几何函数==
:<math>\;_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)^nq^{n\choose 2}\right)^{k-j}z^n.</math>

==关系式==
下列基本超几何函数在q->1时，化为[[超几何函数]]<ref>Roelof KoeKoek, Peter Lesky,Rene Swarttouw,Hypergeometric Orthogonal Polynomials and Their q-Analogues  p15 Springer</ref>

:<math> \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] </math>= <math>\;_{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] </math>

==q二项式定理==

下列公式是[[二项式定理]]的q模拟：

:<math> _1\Phi_{0}([a],[];q;z)=</math><math>\sum_{n=0}^\infty</math><math> \frac{(a;q)_n}{(q;q)_n}</math>

==参考文献==
<references/>
*{{Citation | last1=Fine | first1=Nathan J. | title=Basic hypergeometric series and applications | url=http://www.ams.org/bookstore?fn=20&arg1=survseries&ikey=SURV-27 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1524-3 | mr=956465 | year=1988 | volume=27}}
*{{Citation | last1=Gasper | first1=George | last2=Rahman | first2=Mizan | title=Basic hypergeometric series | publisher=[[Cambridge University Press]] | edition=2nd | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-83357-8 | doi=10.2277/0521833574 | mr=2128719 | year=2004 | volume=96}}
*{{citation|first=Eduard |last=Heine|year=1846|journal= Journal für die reine und angewandte Mathematik|pages=210–212|volume=32|title=Über die Reihe <math>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</math>|url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002145391}}
* [[Eduard Heine]], ''Theorie der Kugelfunctionen'', (1878) ''1'', pp 97–125.
* Eduard Heine, ''Handbuch die Kugelfunctionen. Theorie und Anwendung'' (1898) Springer, Berlin

[[Category:基本超幾何函數]]
{{q超几何函数}}