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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:
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2149ab9c8f41a14aa7ed1538c776dc03fefa4029)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3afcecb4b2cc65942231e217ea54360f857547)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c54c5ea9cd690c70391a61cbd7a828184b7e3831)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f1571d04c9ebc811c407f1f05220aa11776d12)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c54bb9e03fd73b73fa727d9a0223e73125c9f0c)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3ac7b3b72723cf513aec5bc29feafd8b989f3c)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b65247266232aac192a2675c1ddf44c2505ed126)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/741abddcec40afb1a554a6c5887b1eae488bf829)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69ea9e643d06953f0edfd5c8d81b7800a497e6c4)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f9d51229e1f108a17e54d203fa9a3312b845989)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30d248739b47be3e16f7396fd5b114f3bbab80aa)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c02ed1d8f94c2c87599ec0dfaba83c0be1ec60ae)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd46228416452bbbe099b17b45ef8e5864cb8bf5)
![{\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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9533bb35387c992b076d44956f087ea88d91ff6d)
全部20個合流級數如下:
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c34a87b6cfb425f00c20f5cd299cef7abc8bd2)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55e56eb1b591684b2b35d716db8f8f9129299856)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27eaa01a0b7d611f6e0f91e02168e6c1ef71e2cb)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/726280c43daf681e9654734663793bec323b1340)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27779b11edeada02dc66b6bad29ecf5b8e74a0ed)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/426b5aff9a16a9c417861974a3b23b9131a0915d)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f0db4465fcdd2e62a430a394a6f933d012b19c)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/995a9f4d05fcd150c56dc4c8975043b780817117)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee3b57a63a3267db2140271ab4f8d97d3218d2bd)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a268d9f3601b88cfbda3eaaeaf37715296ea51)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69bb99b9b3182aaaa7bb899cc836b97785d084f3)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bb2c5424ae1ce7808f4815dad9ffec70dbb7828)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62037bad8936e4b3da62388c272267962ab735e1)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69035df21f69205d5223fa2eae5932844f4f82fd)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57e8e40ca3656cd81b8f0c4aae7e6ddcfa7fdc13)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b009c5520ea77640680e5fe5f2f362ffb922f418)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/981137019f4dde6844351b767d57c899bd434158)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4893f70342a9c12f10342cd1bfef554384df7f7)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b2ce045a93082c87bc91942708ec858fa7d672)
![{\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!}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0400eedd4b269f4ce53ee591b20fb6edec41b8b)
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。