Horn function

In the theory of special functions in mathematics, the Horn functions (named for Jakob Horn) are the 34 distinct convergent hypergeometric series of order two (i.e. having two independent variables), enumerated by (Horn 1931) (corrected by (Borngässer 1933)). They are listed in (Erdélyi Magnus). B. C. Carlson^[1] revealed a problem with the Horn function classification scheme.^[2] The total 34 Horn functions can be further categorised into 14 complete hypergeometric functions and 20 confluent hypergeometric functions. The complete functions, with their domain of convergence, are:

[math]\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^mw^n}{m!n!}/;|z|\lt 1\land|w|\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;|z|+|w|\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;|z|\lt 1\land|w|\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;\sqrt{|z|}+\sqrt{|w|}\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;|z|+|w|\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;|z|\lt 1\land|w|\lt 1 }[/math]
[math]\displaystyle{ G_3(\alpha,\alpha';z,w)\equiv\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}(\alpha)_{2n-m}(\alpha')_{2m-n}\frac{z^mw^n}{m!n!}/;27|z|^2|w|^2+18|z||w|\pm4(|z|-|w|)\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;4|z||w|+2|w|-|w|^2\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;1/|w|-|z|\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;|z|+|w|^2-|w|\lt 0 }[/math]
[math]\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^mw^n}{m!n!}/;4|z|+2|w|-|w|^2\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;16|z|^2-36|z||w|\pm(8|z|-|w|+27|z||w|^2)\lt -1 }[/math]
[math]\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^mw^n}{m!n!}/;|z||w|^2+|w|\lt 1 }[/math]
[math]\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^mw^n}{m!n!}/;4|z|+2/|s|-1/|s|^2\lt 1 }[/math]

while the confluent functions include:

[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]
[math]\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 !} }[/math]

Notice that some of the complete and confluent functions share the same notation.

References

↑ 'Profile: Bille C. Carlson' in Digital Library of Mathematical Functions. National Institute of Standards and Technology.
↑ Carlson, B. C. (1976). "The need for a new classification of double hypergeometric series". Proc. Amer. Math. Soc. 56: 221–224. doi:10.1090/s0002-9939-1976-0402138-8.

Borngässer, Ludwig (1933), Über hypergeometrische funkionen zweier Veränderlichen, Dissertation, Darmstadt
Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1953), Higher transcendental functions. Vol I, McGraw-Hill Book Company, Inc., New York-Toronto-London, http://apps.nrbook.com/bateman/Vol1.pdf
Horn, J. (1931), "Hypergeometrische Funktionen zweier Veränderlichen", Mathematische Annalen 105 (1): 381–407, doi:10.1007/BF01455825, http://www.digizeitschriften.de/main/dms/img/?IDDOC=364781
J. Horn Math. Ann. 111, 637 (1933)
Srivastava, H. M.; 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

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Original source: https://en.wikipedia.org/wiki/Horn function. Read more

[1] 'Profile: Bille C. Carlson' in Digital Library of Mathematical Functions. National Institute of Standards and Technology.

[2] Carlson, B. C. (1976). "The need for a new classification of double hypergeometric series". Proc. Amer. Math. Soc. 56: 221–224. doi:10.1090/s0002-9939-1976-0402138-8.

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