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 Φ ( 1 ) {\displaystyle \Phi ^{(1)}} q-Appell-4 function1 Φ ( 1 ) ( a ; b , b ′ ; c ; x , y ) = ∑ m , n > 0 {\displaystyle \Phi ^{(1)}(a;b,b';c;x,y)=\sum _{m,n>0}} ( 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 {\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}}}} Φ ( 2 ) ( a ; b , b ′ ; c ; x , y ) = ∑ m , n > 0 {\displaystyle \Phi ^{(2)}(a;b,b';c;x,y)=\sum _{m,n>0}} ( 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 {\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}}}} Φ ( 3 ) ( a , a ′ ; b , b ′ ; c ; x , y ) = ∑ m , n > 0 {\displaystyle \Phi ^{(3)}(a,a';b,b';c;x,y)=\sum _{m,n>0}} ( a , b ; q ) m ∗ ( a ′ , b ′ ; q ) n ∗ x m ∗ y n ( q ; q ) m ∗ ( q ; q ) n ∗ ( c ; q ) m + n {\displaystyle {\frac {(a,b;q)_{m}*(a',b';q)_{n}*x^{m}*y^{n}}{(q;q)_{m}*(q;q)_{n}*(c;q)_{m+n}}}} Φ ( 4 ) ( a ; b ; c , c ′ ; x , y ) = ∑ m , n > 0 {\displaystyle \Phi ^{(4)}(a;b;c,c';x,y)=\sum _{m,n>0}} ( a , b ; q ) m + n ∗ x m ∗ y n ( q , c ; q ) m ∗ ( q , c ′ ; q ) n {\displaystyle {\frac {(a,b;q)_{m+n}*x^{m}*y^{n}}{(q,c;q)_{m}*(q,c';q)_{n}}}} 其中 ( a ; q ) n = ∏ k = 0 n − 1 ( 1 − a q k ) = ( 1 − a ) ( 1 − a q ) ( 1 − a q 2 ) ⋯ ( 1 − a q n − 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})} 为Q阶乘幂 ( a , b ; q ) n = ( a ; q ) n ∗ ( b ; q ) n {\displaystyle (a,b;q)_{n}=(a;q)_{n}*(b;q)_{n}} 关系式 Φ ( 2 ) ( a ; b , b ′ ; c , c ′ ; x , y ) = ( b , a x ; q ) ∞ ( c , x , y ; q ) ∞ ∑ ( ∑ ( a , b ′ ; q ) n ( x ; q ) r ( c / a ; q ) r ( q , c ′ ; q ) n ( q ) r ( a x ; q ) n + r , m = 1.. ∞ ) , r = 1.. ∞ ) {\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 ^ George Gasper, Mizan Rahman,Basic Hypergeometric Series - Page 282,Cambridge University Press,2004 ^ Walled Al-Salam, q-Appell polynomials,Annali di Matematica Pura ed Applicata, 1967 - Springer ^ 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 ^ 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
