Horn函數

Horn函數（以德國數學家雅各布·霍恩命名）是34個不同但都收斂的二階（雙變量）的超幾何級數，由Horn在1931年逐一給出（由Ludwig Borngässer於1933年修正）。34個超幾何級數被進一步分為14個完全的和20個合流的級數，此處「合流」的含義與它在單變量的合流超幾何函數中的含義相同：級數對於任何有限變量都收斂；而「完全」的級數僅對于于單位圓盤內的部分變量收斂。前四個完全的Horn函數即是對應的阿佩爾超幾何函數。全部14個完全的Horn函數，以及它們單位圓盤內的收斂半徑如下：

• $F_{1}(\alpha ;\beta ,\beta ';\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m}(\beta ')_{n}}{(\gamma )_{m+n}}}{\frac {z^{m}w^{n}}{m!n!}}/;|z|<1\land |w|<1$
• $F_{2}(\alpha ;\beta ,\beta ';\gamma ,\gamma ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m}(\beta ')_{n}}{(\gamma )_{m}(\gamma ')_{n}}}{\frac {z^{m}w^{n}}{m!n!}}/;|z|+|w|<1$
• $F_{3}(\alpha ,\alpha ';\beta ,\beta ';\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m}(\alpha ')_{n}(\beta )_{m}(\beta ')_{n}}{(\gamma )_{m+n}}}{\frac {z^{m}w^{n}}{m!n!}}/;|z|<1\land |w|<1$
• $F_{4}(\alpha ;\beta ;\gamma ,\gamma ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m+n}}{(\gamma )_{m}(\gamma ')_{n}}}{\frac {z^{m}w^{n}}{m!n!}}/;{\sqrt {|z|}}+{\sqrt {|w|}}<1$
• $G_{1}(\alpha ;\beta ,\beta ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{m+n}(\beta )_{n-m}(\beta ')_{m-n}{\frac {z^{m}w^{n}}{m!n!}}/;|z|+|w|<1$
• $G_{2}(\alpha ,\alpha ';\beta ,\beta ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{m}(\alpha ')_{n}(\beta )_{n-m}(\beta ')_{m-n}{\frac {z^{m}w^{n}}{m!n!}}/;|z|<1\land |w|<1$
• $G_{3}(\alpha ,\alpha ';z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{2n-m}(\alpha ')_{2m-n}{\frac {z^{m}w^{n}}{m!n!}}/;27|z|^{2}|w|^{2}+18|z||w|\pm 4(|z|-|w|)<1$
• $H_{1}(\alpha ;\beta ;\gamma ;\delta ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m+n}(\gamma )_{n}}{(\delta )_{m}}}{\frac {z^{m}w^{n}}{m!n!}}/;4|z||w|+2|w|-|w|^{2}<1$
• $H_{2}(\alpha ;\beta ;\gamma ;\delta ;\epsilon ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m}(\gamma )_{n}(\delta )_{n}}{(\delta )_{m}}}{\frac {z^{m}w^{n}}{m!n!}}/;1/|w|-|z|<1$
• $H_{3}(\alpha ;\beta ;\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}(\beta )_{n}}{(\gamma )_{m+n}}}{\frac {z^{m}w^{n}}{m!n!}}/;|z|+|w|^{2}-|w|<0$
• $H_{4}(\alpha ;\beta ;\gamma ;\delta ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}(\beta )_{n}}{(\gamma )_{m}(\delta )_{n}}}{\frac {z^{m}w^{n}}{m!n!}}/;4|z|+2|w|-|w|^{2}<1$
• $H_{5}(\alpha ;\beta ;\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}(\beta )_{n-m}}{(\gamma )_{n}}}{\frac {z^{m}w^{n}}{m!n!}}/;16|z|^{2}-36|z||w|\pm (8|z|-|w|+27|z||w|^{2})<-1$
• $H_{6}(\alpha ;\beta ;\gamma ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{2m-n}(\beta )_{n-m}(\gamma )_{n}{\frac {z^{m}w^{n}}{m!n!}}/;|z||w|^{2}+|w|<1$
• $H_{7}(\alpha ;\beta ;\gamma ;\delta ;z,w)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m-n}(\beta )_{n}(\gamma )_{n}}{(\delta )_{m}}}{\frac {z^{m}w^{n}}{m!n!}}/;4|z|+2/|s|-1/|s|^{2}<1$

全部20個合流級數如下：

• $\Phi _{1}\left(\alpha ;\beta ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $\Phi _{2}\left(\beta ,\beta ';\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\beta )_{m}(\beta ')_{n}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $\Phi _{3}\left(\beta ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\beta )_{m}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $\Psi _{1}\left(\alpha ;\beta ;\gamma ,\gamma ';x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}(\beta )_{m}}{(\gamma )_{m}(\gamma ')_{n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $\Psi _{2}\left(\alpha ;\gamma ,\gamma ';x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m+n}}{(\gamma )_{m}(\gamma ')_{n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $\Xi _{1}\left(\alpha ,\alpha ';\beta ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m}(\alpha ')_{n}(\beta )_{m}}{(\gamma )_{m+n}(\gamma ')_{n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $\Xi _{2}\left(\alpha ;\beta ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m}(\alpha )_{m}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $\Gamma _{1}\left(\alpha ;\beta ,\beta ';x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{m}(\beta )_{n-m}(\beta ')_{m-n}{\frac {x^{m}y^{n}}{m!n!}}$
• $\Gamma _{2}\left(\beta ,\beta ';x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\beta )_{n-m}(\beta ')_{m-n}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{1}\left(\alpha ;\beta ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m+n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{2}\left(\alpha ;\beta ;\gamma ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m}(\gamma )_{n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{3}\left(\alpha ;\beta ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{m}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{4}\left(\alpha ;\gamma ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\gamma )_{n}}{(\delta )_{n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{5}\left(\alpha ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{6}\left(\alpha ;\gamma ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}}{(\gamma )_{m+n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{7}\left(\alpha ;\gamma ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m+n}}{(\gamma )_{m}(\delta )_{n}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{8}\left(\alpha ;\beta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }(\alpha )_{2m-n}(\beta )_{n-m}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{9}\left(\alpha ;\beta ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m-n}(\beta )_{n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{10}\left(\alpha ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{2m-n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n!}}$
• $H_{11}\left(\alpha ;\beta ;\gamma ;\delta ;x,y\right)\equiv \sum _{m=0}^{\infty }\sum _{n=0}^{\infty }{\frac {(\alpha )_{m-n}(\beta )_{n}(\gamma )_{n}}{(\delta )_{m}}}{\frac {x^{m}y^{n}}{m!n!}}$

注意部分完全級數和合流級數的記號相同。全部Horn函數都是Kampé de Fériet函數的特例。