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 {{harvtxt|Horn|1931}} (corrected by {{harvtxt|Borngässer|1933}}). They are listed in {{harv|Erdélyi|Magnus|Oberhettinger|Tricomi|1953|loc=section 5.7.1}}. B. C. Carlson[http://dlmf.nist.gov/about/bio/BCCarlson 'Profile: Bille C. Carlson' in Digital Library of Mathematical Functions. National Institute of Standards and Technology.] revealed a problem with the Horn function classification scheme.

{{cite journal|author=Carlson, B. C.|title=The need for a new classification of double hypergeometric series|journal=Proc. Amer. Math. Soc.|year=1976|volume=56|pages=221–224|mr=0402138|doi=10.1090/s0002-9939-1976-0402138-8|doi-access=free}}

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:

• $$

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|<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^mw^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^mw^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^mw^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)_{n-m}(\beta')_{m-n}\frac{z^mw^n}{m!n!}/;|z|+|w|<1

• $$

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|<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^mw^n}{m!n!}/;27|z|^2|w|^2+18|z||w|\pm4(|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^mw^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^mw^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^mw^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^mw^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^mw^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^mw^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^mw^n}{m!n!}/;4|z|+2/|s|-1/|s|^2<1

while the confluent functions include:

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

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