Horn函数

Template:Multiple issues Horn函数（以德国数学家Template:Link-en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：

• $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m+n}(\beta {)}_m({\beta }^{\prime }{)}_n}{(\gamma {)}_{m+n}}\frac{z^m{w}^n}{m!n!}/;|z|<1\wedge |w|<1$
• $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m+n}(\beta {)}_m({\beta }^{\prime }{)}_n}{(\gamma {)}_m({\gamma }^{\prime }{)}_n}\frac{z^m{w}^n}{m!n!}/;|z|+|w|<1$
• $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_m({\alpha }^{\prime }{)}_n(\beta {)}_m({\beta }^{\prime }{)}_n}{(\gamma {)}_{m+n}}\frac{z^m{w}^n}{m!n!}/;|z|<1\wedge |w|<1$
• $F_4(\alpha ;\beta ;\gamma ,{\gamma }^{\prime };z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m+n}(\beta {)}_{m+n}}{(\gamma {)}_m({\gamma }^{\prime }{)}_n}\frac{z^m{w}^n}{m!n!}/;\sqrt{|z|}+\sqrt{|w|}<1$
• $G_1(\alpha ;\beta ,{\beta }^{\prime };z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}(\alpha {)}_{m+n}(\beta {)}_{n-m}({\beta }^{\prime }{)}_{m-n}\frac{z^m{w}^n}{m!n!}/;|z|+|w|<1$
• $G_2(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}(\alpha {)}_m({\alpha }^{\prime }{)}_n(\beta {)}_{n-m}({\beta }^{\prime }{)}_{m-n}\frac{z^m{w}^n}{m!n!}/;|z|<1\wedge |w|<1$
• $G_3(\alpha ,{\alpha }^{\prime };z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}(\alpha {)}_{2n-m}({\alpha }^{\prime }{)}_{2m-n}\frac{z^m{w}^n}{m!n!}/;27|z{|}^2|w{|}^2+18|z||w|\pm 4(|z|-|w|)<1$
• $H_1(\alpha ;\beta ;\gamma ;\delta ;z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m-n}(\beta {)}_{m+n}(\gamma {)}_n}{(\delta {)}_m}\frac{z^m{w}^n}{m!n!}/;4|z||w|+2|w|-|w{|}^2<1$
• $H_2(\alpha ;\beta ;\gamma ;\delta ;ϵ;z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m-n}(\beta {)}_m(\gamma {)}_n(\delta {)}_n}{(\delta {)}_m}\frac{z^m{w}^n}{m!n!}/;1/|w|-|z|<1$
• $H_3(\alpha ;\beta ;\gamma ;z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{2m+n}(\beta {)}_n}{(\gamma {)}_{m+n}}\frac{z^m{w}^n}{m!n!}/;|z|+|w{|}^2-|w|<0$
• $H_4(\alpha ;\beta ;\gamma ;\delta ;z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{2m+n}(\beta {)}_n}{(\gamma {)}_m(\delta {)}_n}\frac{z^m{w}^n}{m!n!}/;4|z|+2|w|-|w{|}^2<1$
• $H_5(\alpha ;\beta ;\gamma ;z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{2m+n}(\beta {)}_{n-m}}{(\gamma {)}_n}\frac{z^m{w}^n}{m!n!}/;16|z{|}^2-36|z||w|\pm (8|z|-|w|+27|z||w{|}^2)<-1$
• $H_6(\alpha ;\beta ;\gamma ;z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}(\alpha {)}_{2m-n}(\beta {)}_{n-m}(\gamma {)}_n\frac{z^m{w}^n}{m!n!}/;|z||w{|}^2+|w|<1$
• $H_7(\alpha ;\beta ;\gamma ;\delta ;z,w)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{2m-n}(\beta {)}_n(\gamma {)}_n}{(\delta {)}_m}\frac{z^m{w}^n}{m!n!}/;4|z|+2/|s|-1/|s{|}^2<1$

全部20个合流级数如下：

• ${\mathrm{\Phi }}_1(\alpha ;\beta ;\gamma ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m+n}(\beta {)}_m}{(\gamma {)}_{m+n}}\frac{x^m{y}^n}{m!n!}$
• ${\mathrm{\Phi }}_2(\beta ,{\beta }^{\prime };\gamma ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\beta {)}_m({\beta }^{\prime }{)}_n}{(\gamma {)}_{m+n}}\frac{x^m{y}^n}{m!n!}$
• ${\mathrm{\Phi }}_3(\beta ;\gamma ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\beta {)}_m}{(\gamma {)}_{m+n}}\frac{x^m{y}^n}{m!n!}$
• ${\mathrm{\Psi }}_1(\alpha ;\beta ;\gamma ,{\gamma }^{\prime };x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m+n}(\beta {)}_m}{(\gamma {)}_m({\gamma }^{\prime }{)}_n}\frac{x^m{y}^n}{m!n!}$
• ${\mathrm{\Psi }}_2(\alpha ;\gamma ,{\gamma }^{\prime };x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m+n}}{(\gamma {)}_m({\gamma }^{\prime }{)}_n}\frac{x^m{y}^n}{m!n!}$
• ${\mathrm{\Xi }}_1(\alpha ,{\alpha }^{\prime };\beta ;\gamma ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_m({\alpha }^{\prime }{)}_n(\beta {)}_m}{(\gamma {)}_{m+n}({\gamma }^{\prime }{)}_n}\frac{x^m{y}^n}{m!n!}$
• ${\mathrm{\Xi }}_2(\alpha ;\beta ;\gamma ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_m(\alpha {)}_m}{(\gamma {)}_{m+n}}\frac{x^m{y}^n}{m!n!}$
• ${\mathrm{\Gamma }}_1(\alpha ;\beta ,{\beta }^{\prime };x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}(\alpha {)}_m(\beta {)}_{n-m}({\beta }^{\prime }{)}_{m-n}\frac{x^m{y}^n}{m!n!}$
• ${\mathrm{\Gamma }}_2(\beta ,{\beta }^{\prime };x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}(\beta {)}_{n-m}({\beta }^{\prime }{)}_{m-n}\frac{x^m{y}^n}{m!n!}$
• $H_1(\alpha ;\beta ;\delta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m-n}(\beta {)}_{m+n}}{(\delta {)}_m}\frac{x^m{y}^n}{m!n!}$
• $H_2(\alpha ;\beta ;\gamma ;\delta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m-n}(\beta {)}_m(\gamma {)}_n}{(\delta {)}_m}\frac{x^m{y}^n}{m!n!}$
• $H_3(\alpha ;\beta ;\delta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m-n}(\beta {)}_m}{(\delta {)}_m}\frac{x^m{y}^n}{m!n!}$
• $H_4(\alpha ;\gamma ;\delta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m-n}(\gamma {)}_n}{(\delta {)}_n}\frac{x^m{y}^n}{m!n!}$
• $H_5(\alpha ;\delta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m-n}}{(\delta {)}_m}\frac{x^m{y}^n}{m!n!}$
• $H_6(\alpha ;\gamma ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{2m+n}}{(\gamma {)}_{m+n}}\frac{x^m{y}^n}{m!n!}$
• $H_7(\alpha ;\gamma ;\delta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{2m+n}}{(\gamma {)}_m(\delta {)}_n}\frac{x^m{y}^n}{m!n!}$
• $H_8(\alpha ;\beta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}(\alpha {)}_{2m-n}(\beta {)}_{n-m}\frac{x^m{y}^n}{m!n!}$
• $H_9(\alpha ;\beta ;\delta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{2m-n}(\beta {)}_n}{(\delta {)}_m}\frac{x^m{y}^n}{m!n!}$
• $H_{10}(\alpha ;\delta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{2m-n}}{(\delta {)}_m}\frac{x^m{y}^n}{m!n!}$
• $H_{11}(\alpha ;\beta ;\gamma ;\delta ;x,y)\equiv \sum _{m=0}^{\mathrm{\infty }}\sum _{n=0}^{\mathrm{\infty }}\frac{(\alpha {)}_{m-n}(\beta {)}_n(\gamma {)}_n}{(\delta {)}_m}\frac{x^m{y}^n}{m!n!}$

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。