q阿佩爾函數(q-Appell function)又名q阿佩爾多項式(q-Appell polynomials)是數學家Jackson創立的阿佩爾函數的q模擬[1][2]
《美國國家標準局數學函數手冊》中給出的定義如下[3]
q-阿佩爾函數是二變數超幾何函數,共四個:
Q Appell function
q-Appell-4 function1
![{\displaystyle \Phi ^{(1)}(a;b,b';c;x,y)=\sum _{m,n>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95c225ffd73c0d0cf93474771aeb8fbc3046473c)
![{\displaystyle \Phi ^{(2)}(a;b,b';c;x,y)=\sum _{m,n>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b281ba1c10fdef916b61a3d72550fd5c569e5893)
![{\displaystyle \Phi ^{(3)}(a,a';b,b';c;x,y)=\sum _{m,n>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34f8d6c2bb35616b87ba30c7f2165310a6c628c3)
![{\displaystyle \Phi ^{(4)}(a;b;c,c';x,y)=\sum _{m,n>0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4acbaf325bbebd5e3041807f536116b5dfe9f5)
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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b46d2065b6578f314f0353ee519c16683dd948b4)
為Q階乘冪
![{\displaystyle (a,b;q)_{n}=(a;q)_{n}*(b;q)_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b724acca80a0721e8d8460e2cb2b5eaffef62cc9)
關係式[編輯]
[4].
參考文獻[編輯]
- ^ 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,《美國國家標準局 數學函數手冊》 NIST Handbook of Mathematical Functions, p423,p936 劍橋大學出版社 Cambridge University Press, 2010
- ^ Oliver,《美國國家標準局 數學函數手冊》 NIST Handbook of Mathematical Functions, p430 劍橋大學出版社 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(頁面存檔備份,存於網際網路檔案館)
- On Models of Uq(sl(2)) and q-Appell Functions Using a q-Integral Transformation