Función de bocina

En la teoría de funciones especiales en matemáticas , las funciones de Horn (nombradas así por Jakob Horn ) son las 34 series hipergeométricas convergentes distintas de orden dos (es decir, que tienen 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 funciones de Horn. ^[2] Las 34 funciones de Horn en total se pueden categorizar en 14 funciones hipergeométricas completas y 20 funciones hipergeométricas confluentes. Las funciones completas, con su dominio de convergencia, son:

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

Mientras que las funciones confluentes incluyen:

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

Tenga en cuenta que algunas de las funciones completas y confluentes comparten la misma notación.

Referencias

^ 'Perfil: Bille C. Carlson' en Biblioteca digital de funciones matemáticas. Instituto Nacional de Estándares y Tecnología.
^ Carlson, BC (1976). "La necesidad de una nueva clasificación de series hipergeométricas dobles". Proc. Amer. Math. Soc . 56 : 221–224. doi : 10.1090/s0002-9939-1976-0402138-8 . MR 0402138.

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 desde el original (PDF) el 2011-08-11 , consultado el 2015-08-23
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), Series hipergeométricas gaussianas múltiples , Serie Ellis Horwood: Matemáticas y sus aplicaciones, Chichester: Ellis Horwood Ltd., ISBN 978-0-85312-602-7, Sr. 0834385