Horn函数

Horn函数（以德国数学家雅各布·霍恩（英语：Jakob Horn）命名）是34个不同但都收敛的二阶（双变量）的超几何级数，由Horn在1931年逐一给出（由Ludwig Borngässer于1933年修正）。34个超几何级数被进一步分为14个完全的和20个合流的级数，此处“合流”的含义与它在单变量的合流超几何函数中的含义相同：级数对于任何有限变量都收敛；而“完全”的级数仅对于于单位圆盘内的部分变量收敛。前四个完全的Horn函数即是对应的阿佩尔超几何函数。全部14个完全的Horn函数，以及它们单位圆盘内的收敛半径如下： F 1 ( α ; β , β ′ ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m ( β ′ ) n ( γ ) m + n z m w n m ! n ! / ; | z | < 1 ∧ | w | < 1 {\displaystyle 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 ( α ; β , β ′ ; γ , γ ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m ( β ′ ) n ( γ ) m ( γ ′ ) n z m w n m ! n ! / ; | z | + | w | < 1 {\displaystyle 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 ( α , α ′ ; β , β ′ ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( α ′ ) n ( β ) m ( β ′ ) n ( γ ) m + n z m w n m ! n ! / ; | z | < 1 ∧ | w | < 1 {\displaystyle 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 ( α ; β ; γ , γ ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m + n ( γ ) m ( γ ′ ) n z m w n m ! n ! / ; | z | + | w | < 1 {\displaystyle 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 ( α ; β , β ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) n − m ( β ′ ) m − n z m w n m ! n ! / ; | z | + | w | < 1 {\displaystyle 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 ( α , α ′ ; β , β ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( α ′ ) n ( β ) n − m ( β ′ ) m − n z m w n m ! n ! / ; | z | < 1 ∧ | w | < 1 {\displaystyle 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 ( α , α ′ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 n − m ( α ′ ) 2 m − n z m w n m ! n ! / ; 27 | z | 2 | w | 2 + 18 | z | | w | ± 4 ( | z | − | w | ) < 1 {\displaystyle 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 ( α ; β ; γ ; δ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m + n ( γ ) n ( δ ) m z m w n m ! n ! / ; 4 | z | | w | + 2 | w | − | w | 2 < 1 {\displaystyle 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 ( α ; β ; γ ; δ ; ϵ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m ( γ ) n ( δ ) n ( δ ) m z m w n m ! n ! / ; 1 / | w | − | z | < 1 {\displaystyle 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 ( α ; β ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( β ) n ( γ ) m + n z m w n m ! n ! / ; | z | + | w | 2 − | w | < 0 {\displaystyle 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 ( α ; β ; γ ; δ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( β ) n ( γ ) m ( δ ) n z m w n m ! n ! / ; 4 | z | + 2 | w | − | w | 2 < 1 {\displaystyle 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 ( α ; β ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( β ) n − m ( γ ) n z m w n m ! n ! / ; 16 | z | 2 − 36 | z | | w | ± ( 8 | z | − | w | + 27 | z | | w | 2 ) < − 1 {\displaystyle 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 ( α ; β ; γ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( β ) n − m ( γ ) n z m w n m ! n ! / ; | z | | w | 2 + | w | < 1 {\displaystyle 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 ( α ; β ; γ ; δ ; z , w ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( β ) n ( γ ) n ( δ ) m z m w n m ! n ! / ; 4 | z | + 2 / | s | − 1 / | s | 2 < 1 {\displaystyle 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个合流级数如下： Φ 1 ( α ; β ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m ( γ ) m + n x m y n m ! n ! {\displaystyle \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!}}} Φ 2 ( β , β ′ ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( β ) m ( β ′ ) n ( γ ) m + n x m y n m ! n ! {\displaystyle \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!}}} Φ 3 ( β ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( β ) m ( γ ) m + n x m y n m ! n ! {\displaystyle \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!}}} Ψ 1 ( α ; β ; γ , γ ′ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( β ) m ( γ ) m ( γ ′ ) n x m y n m ! n ! {\displaystyle \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!}}} Ψ 2 ( α ; γ , γ ′ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m + n ( γ ) m ( γ ′ ) n x m y n m ! n ! {\displaystyle \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!}}} Ξ 1 ( α , α ′ ; β ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( α ′ ) n ( β ) m ( γ ) m + n ( γ ′ ) n x m y n m ! n ! {\displaystyle \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!}}} Ξ 2 ( α ; β ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( α ) m ( γ ) m + n x m y n m ! n ! {\displaystyle \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!}}} Γ 1 ( α ; β , β ′ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m ( β ) n − m ( β ′ ) m − n x m y n m ! n ! {\displaystyle \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!}}} Γ 2 ( β , β ′ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( β ) n − m ( β ′ ) m − n x m y n m ! n ! {\displaystyle \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 ( α ; β ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m + n ( δ ) m x m y n m ! n ! {\displaystyle 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 ( α ; β ; γ ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m ( γ ) n ( δ ) m x m y n m ! n ! {\displaystyle 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 ( α ; β ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) m ( δ ) m x m y n m ! n ! {\displaystyle 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 ( α ; γ ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( γ ) n ( δ ) n x m y n m ! n ! {\displaystyle 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 ( α ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( δ ) m x m y n m ! n ! {\displaystyle 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 ( α ; γ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( γ ) m + n x m y n m ! n ! {\displaystyle 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 ( α ; γ ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m + n ( γ ) m ( δ ) n x m y n m ! n ! {\displaystyle 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 ( α ; β ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( β ) n − m x m y n m ! n ! {\displaystyle 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 ( α ; β ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( β ) n ( δ ) m x m y n m ! n ! {\displaystyle 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 ( α ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) 2 m − n ( δ ) m x m y n m ! n ! {\displaystyle 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 ( α ; β ; γ ; δ ; x , y ) ≡ ∑ m = 0 ∞ ∑ n = 0 ∞ ( α ) m − n ( β ) n ( γ ) n ( δ ) m x m y n m ! n ! {\displaystyle 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!}}} 注意部分完全级数和合流级数的记号相同。全部Horn函数都是Kampé de Fériet函数的特例。