Q阿佩尔函数

q阿佩尔函数（q-Appell function）又名q阿佩尔多项式（q-Appell polynomials）是数学家Jackson创立的阿佩尔函数的q模拟^[1][2]

《美国国家标准局数学函数手册》中给出的定义如下^[3]

q-阿佩尔函数是二变数超几何函数，共四个：

Q Appell function ${\displaystyle \Phi ^{(1)}}$

q-Appell-4 function1

${\displaystyle \Phi ^{(1)}(a;b,b';c;x,y)=\sum _{m,n>0}}$${\displaystyle {\frac {(a;q)_{m+n}*(b;q)_{m}*(b';q)_{n}*x^{m}*y^{n}}{(q;q)_{m}*(q;q)_{n}*(c;q)_{m+n}}}}$

${\displaystyle \Phi ^{(2)}(a;b,b';c;x,y)=\sum _{m,n>0}}$${\displaystyle {\frac {(a;q)_{m+n}*(b;q)_{m}*(b';q)_{n}*x^{m}*y^{n}}{(q;q)_{m}*(q;q)_{n}*(c;q)_{m+n}}}}$

${\displaystyle \Phi ^{(3)}(a,a';b,b';c;x,y)=\sum _{m,n>0}}$${\displaystyle {\frac {(a,b;q)_{m}*(a',b';q)_{n}*x^{m}*y^{n}}{(q;q)_{m}*(q;q)_{n}*(c;q)_{m+n}}}}$

${\displaystyle \Phi ^{(4)}(a;b;c,c';x,y)=\sum _{m,n>0}}$${\displaystyle {\frac {(a,b;q)_{m+n}*x^{m}*y^{n}}{(q,c;q)_{m}*(q,c';q)_{n}}}}$

其中

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

为Q阶乘幂

${\displaystyle (a,b;q)_{n}=(a;q)_{n}*(b;q)_{n}}$

关系式

${\displaystyle \Phi ^{(2)}(a;b,b';c,c';x,y)={\frac {(b,ax;q)_{\infty }}{(c,x,y;q)_{\infty }}}\sum (\sum {\frac {(a,b';q)_{n}(x;q)_{r}(c/a;q)_{r}}{(q,c';q)_{n}(q)_{r}(ax;q)_{n+r}}},m=1..\infty ),r=1..\infty )}$  ^[4].

参考文献

1. George Gasper, Mizan Rahman,Basic Hypergeometric Series - Page 282,Cambridge University Press,2004
2. Walled Al-Salam, q-Appell polynomials,Annali di Matematica Pura ed Applicata, 1967 - Springer
3. Oliver,《美国国家标准局 数学函数手册》 NIST Handbook of Mathematical Functions, p423,p936 剑桥大学出版社 Cambridge University Press, 2010
4. Oliver,《美国国家标准局 数学函数手册》 NIST Handbook of Mathematical Functions, p430 剑桥大学出版社 Cambridge University Press, 2010

• H. M. Srivastava,Some Characterizations of Appell and Q-Appell Polynomials,University of Victoria, Department of Mathematics, 1981
• Thomas Ernst,Convergence Aspects for Q-appell Functions,Uppsala universitet, 2010
• Thomas Ernst,A Comprehensive Treatment of Q-Calculus,p381,p432,Birkhaus 2012
• Basic Hypergeometric Series（页面存档备份，存于互联网档案馆）
• On Models of Uq(sl(2)) and q-Appell Functions Using a q-Integral Transformation