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 (corrected by ). They are listed in. B. C. Carlson revealed a problem with the Horn function classification scheme. 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: Notice that some of the complete and confluent functions share the same notation.
 * $$\Phi_{1}\left(\alpha;\beta;\gamma;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m+n}(\beta)_{m}}{(\gamma)_{m+n}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$\Phi_{2}\left(\beta,\beta';\gamma;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\beta)_{m}(\beta')_{n}}{(\gamma)_{m+n}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$\Phi_{3}\left(\beta;\gamma;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\beta)_{m}}{(\gamma)_{m+n}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$\Psi_{1}\left(\alpha;\beta;\gamma,\gamma';x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m+n}(\beta)_{m}}{(\gamma)_{m}(\gamma')_{n}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$\Psi_{2}\left(\alpha;\gamma,\gamma';x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m+n}}{(\gamma)_{m}(\gamma')_{n}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$\Xi_{1}\left(\alpha,\alpha';\beta;\gamma;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m}(\alpha')_{n}(\beta)_m}{(\gamma)_{m+n}(\gamma')_{n}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$\Xi_{2}\left(\alpha;\beta;\gamma;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m}(\alpha)_{m}}{(\gamma)_{m+n}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$\Gamma_{1}\left(\alpha;\beta,\beta';x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} (\alpha)_m (\beta)_{n-m}(\beta')_{m-n}\frac{x^{m} y^{n}}{m ! n !}$$
 * $$\Gamma_{2}\left(\beta,\beta';x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}(\beta)_{n-m}(\beta')_{m-n}\frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{1}\left(\alpha;\beta;\delta ;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m-n}(\beta)_{m+n}}{(\delta)_m} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{2}\left(\alpha;\beta;\gamma;\delta;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m-n}(\beta)_{m}(\gamma)_n}{(\delta)_m} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{3}\left(\alpha;\beta;\delta;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m-n}(\beta)_{m}}{(\delta)_{m}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{4}\left(\alpha;\gamma;\delta ;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m-n}(\gamma)_{n}}{(\delta)_n} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{5}\left(\alpha;\delta;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{m-n}}{(\delta)_m} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{6}\left(\alpha;\gamma;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{(\alpha)_{2m+n}}{(\gamma)_{m+n}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{7}\left(\alpha;\gamma;\delta;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{(\alpha)_{2m+n}}{(\gamma)_m(\delta)_n} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{8}\left(\alpha;\beta;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} (\alpha)_{2m-n}(\beta)_{n-m} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{9}\left(\alpha;\beta;\delta;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(\alpha)_{2m-n}(\beta)_{n}}{(\delta)_m} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{10}\left(\alpha;\delta;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{(\alpha)_{2m-n}}{(\delta)_{m}} \frac{x^{m} y^{n}}{m ! n !}$$
 * $$H_{11}\left(\alpha;\beta;\gamma;\delta;x, y\right)\equiv\sum_{m=0}^{\infty} \sum_{n=0}^{\infty}\frac{(\alpha)_{m-n}(\beta)_n(\gamma)_n}{(\delta)_m} \frac{x^{m} y^{n}}{m ! n !}$$