Función de bocina

En la teoría de funciones especiales en matemáticas , las funciones de Horn (llamadas así por Jakob Horn ) son las 34 series hipergeométricas convergentes distintas de orden dos (es decir, con dos variables independientes), enumeradas por Horn (1931) (corregidas por Borngässer (1933) ). Se enumeran en ( Erdélyi et al. 1953 , sección 5.7.1) . BC Carlson ^{[ 1 ]} reveló un problema con el esquema de clasificación de las funciones de Horn. ^{[ 2 ]} Las 34 funciones de Horn se pueden categorizar además en 14 funciones hipergeométricas completas y 20 funciones hipergeométricas confluentes. Las funciones completas, con su dominio de convergencia, son:

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

mientras que las funciones confluentes incluyen:

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

Nótese que algunas de las funciones completas y confluentes comparten la misma notación.

• Borngässer, Ludwig (1933), Über hypergeometrische funkionen zweier Veränderlichen , Disertación, Darmstadt
• Erdélyi, Arthur; Magnus, Wilhelm ; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Funciones trascendentales superiores. Vol. I (PDF) , McGraw-Hill Book Company, Inc., Nueva York-Toronto-Londres, MR 0058756 , archivado del original (PDF) el 11 de agosto de 2011 , consultado el 23 de agosto de 2015. 
• Horn, J. (1931), "Hypergeometrische Funktionen zweier Veränderlichen" , Mathematische Annalen , 105 (1): 381– 407, doi : 10.1007/BF01455825 , S2CID 179177588 
• J. Horn Math. Ann. 111 , 637 (1933)
• Srivastava, HM; Karlsson, Per W. (1985), Multiple Gaussian hypergeometric series , Ellis Horwood Series: Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-85312-602-7, MR 0834385