AppellF2 - Maple Help

AppellF2

The AppellF2 function

 Calling Sequence AppellF2($a,{b}_{1},{b}_{2},{c}_{1},{c}_{2},{z}_{1},{z}_{2}$)

Parameters

 $a$ - algebraic expression ${b}_{1}$ - algebraic expression ${b}_{2}$ - algebraic expression ${c}_{1}$ - algebraic expression ${c}_{2}$ - algebraic expression ${z}_{1}$ - algebraic expression ${z}_{2}$ - algebraic expression

Description

 • As is the case of all the four multi-parameter Appell functions, AppellF2, is a doubly hypergeometric function that includes as particular cases the 2F1 hypergeometric and some cases of the MeijerG function, and with them most of the known functions of mathematical physics. Among other situations, AppellF2 appears in the solution to differential equations in general relativity, quantum mechanics, and molecular and atomic physics.
 Initialization: Set the display of special functions in output to typeset mathematical notation (textbook notation):
 > $\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right):$
 The definition of the AppellF2 series and the corresponding domain of convergence can be seen through the FunctionAdvisor
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{AppellF2}\right)$
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left(\mathrm{c__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{c__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\left|\mathrm{z__1}\right|{+}\left|\mathrm{z__2}\right|{<}{1}\right]$ (1)
 A distinction is made between the AppellF2 doubly hypergeometric series, with the restricted domain of convergence shown above, and the AppellF2 function, that coincides with the series in its domain of convergence but also extends it analytically to the whole complex plane.
 From the definition above, by swapping the AppellF2 variables subscripted with the numbers 1 and 2, the function remains the same; hence
 > $\mathrm{FunctionAdvisor}\left(\mathrm{symmetries},\mathrm{AppellF2}\right)$
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}\mathrm{c__2}{,}\mathrm{c__1}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right]$ (2)
 From the series' definition, AppellF2 is singular (division by zero) when the ${c}_{1}$ and/or ${c}_{2}$ parameters entering the pochhammer functions in the denominator of the series are non-positive integers, because these pochhammer functions will be equal to zero when the summation index of the series is bigger than the absolute value of the corresponding ${c}_{1}$ or ${c}_{2}$ parameter.
 For an analogous reason, when the $a$ and/or both ${b}_{1}$ and ${b}_{2}$ parameters entering the pochhammer functions in the numerator of the series are non-positive integers, the series will truncate and AppellF2 will be polynomial. As is the case of the hypergeometric function, when the pochhammers in both the numerator and the denominator have non-positive integer arguments, AppellF2 is polynomial if the absolute value of the non-positive integers in the pochhammers of the numerator are smaller than or equal to the absolute value of the non-positive integer (parameters ${c}_{1},{c}_{2}$) in the pochhammers in the denominator, and singular otherwise. Consult the FunctionAdvisor for comprehensive information on the combinations of all these conditions. For example, the singular cases happen when either the following conditions hold
 > $\mathrm{FunctionAdvisor}\left(\mathrm{singularities},\mathrm{AppellF2}\right)$
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\left(\mathrm{c__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left(\mathrm{c__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{a}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{<}\mathrm{c__1}{\wedge }\mathrm{b__1}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left(\mathrm{c__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{a}{<}\mathrm{c__2}\right){\vee }\left(\mathrm{c__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{<}\mathrm{c__2}\right){\vee }\left(\mathrm{c__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{<}\mathrm{c__2}{\wedge }\mathrm{b__2}{<}\mathrm{c__2}\right)\right]$ (3)
 The AppellF2 series is analytically extended to the AppellF2 function defined over the whole complex plane using identities and mainly by integral representations in terms of Eulerian integrals:
 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral_form},\mathrm{AppellF2}\right)$
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__1}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{{u}}^{\mathrm{b__1}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}{-}\frac{\mathrm{z__2}}{{u}{}\mathrm{z__1}{-}{1}}\right)}{{\left({1}{-}{u}\right)}^{{-}\mathrm{c__1}{+}\mathrm{b__1}{+}{1}}{}{\left({-}{u}{}\mathrm{z__1}{+}{1}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__1}{-}\mathrm{b__1}\right)}{,}\mathrm{z__1}{\ne }{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__1}{+}\mathrm{b__1}\right)\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{{u}}^{\mathrm{b__2}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}{-}\frac{\mathrm{z__1}}{\mathrm{z__2}{}{u}{-}{1}}\right)}{{\left({1}{-}{u}\right)}^{{1}{+}\mathrm{b__2}{-}\mathrm{c__2}}{}{\left({-}\mathrm{z__2}{}{u}{+}{1}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}\mathrm{b__2}\right)}{,}\mathrm{z__2}{\ne }{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__2}{+}\mathrm{b__2}\right)\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}\left({{\int }}_{{0}}^{{1}}{{\int }}_{{0}}^{{1}}\frac{{{u}}^{\mathrm{b__1}{-}{1}}{}{{v}}^{\mathrm{b__2}{-}{1}}}{{\left({1}{-}{u}\right)}^{{-}\mathrm{c__1}{+}\mathrm{b__1}{+}{1}}{}{\left({1}{-}{v}\right)}^{{1}{+}\mathrm{b__2}{-}\mathrm{c__2}}{}{\left({-}{u}{}\mathrm{z__1}{-}{v}{}\mathrm{z__2}{+}{1}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{v}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__1}{-}\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}\mathrm{b__2}\right)}{,}{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__1}{+}\mathrm{b__1}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__2}{+}\mathrm{b__2}\right)\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{\int }}_{{0}}^{{\mathrm{\infty }}}\frac{{{u}}^{{a}{-}{1}}{}{}_{{1}}{F}_{{1}}{}\left(\mathrm{b__1}{;}\mathrm{c__1}{;}{u}{}\mathrm{z__1}\right){}{}_{{1}}{F}_{{1}}{}\left(\mathrm{b__2}{;}\mathrm{c__2}{;}\mathrm{z__2}{}{u}\right)}{{{ⅇ}}^{{u}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}}{{\mathrm{\Gamma }}{}\left({a}\right)}{,}{\mathrm{\Re }}{}\left(\mathrm{z__1}{+}\mathrm{z__2}\right){<}{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left({a}\right)\right]$ (4)
 These integral representations are also the starting point for the derivation of many of the identities known for AppellF2.
 AppellF2 also satisfies a linear system of partial differential equations of second order
 > $\mathrm{FunctionAdvisor}\left(\mathrm{DE},\mathrm{AppellF2}\right)$
 $\left[{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\left[\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__1}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{-}\frac{\mathrm{z__2}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }\mathrm{z__1}{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{-}{1}}{+}\frac{\left(\left({-}{a}{-}\mathrm{b__1}{-}{1}\right){}\mathrm{z__1}{+}\mathrm{c__1}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{-}\frac{\mathrm{b__1}{}\mathrm{z__2}{}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{-}\frac{{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}{}\mathrm{b__1}}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{,}\frac{{{\partial }}^{{2}}}{{\partial }\mathrm{z__1}{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\left({1}{-}\mathrm{z__2}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__2}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}}{-}\frac{\mathrm{b__2}{}\left(\frac{{\partial }}{{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__2}}{+}\frac{\left(\left({-}{a}{-}\mathrm{b__2}{-}{1}\right){}\mathrm{z__2}{+}\mathrm{c__2}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\mathrm{z__2}}{-}\frac{{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}{}\mathrm{b__2}}{\mathrm{z__1}{}\mathrm{z__2}}\right]\right]$ (5)

Examples

 Initialization: Set the display of special functions in output to typeset mathematical notation (textbook notation):
 > $\mathrm{Typesetting}:-\mathrm{EnableTypesetRule}\left(\mathrm{Typesetting}:-\mathrm{SpecialFunctionRules}\right):$

The conditions for both the singular and the polynomial cases can also be seen from the AppellF2. For example, the fourteen polynomial cases of AppellF2 are

 > $\mathrm{AppellF2}:-\mathrm{SpecialValues}:-\mathrm{Polynomial}\left(\right)$
 ${14}{,}\left({a}{,}{\mathrm{b1}}{,}{\mathrm{b2}}{,}{\mathrm{c1}}{,}{\mathrm{c2}}{,}{\mathrm{z1}}{,}{\mathrm{z2}}\right){↦}'\left[\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{\le }{a}{,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{\le }{a}\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{\le }{a}{,}{\mathrm{c2}}{\le }{a}\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{\le }{\mathrm{b1}}{,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{\le }{\mathrm{b2}}\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{\le }{\mathrm{b1}}{,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{\le }{\mathrm{b2}}\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{\le }{\mathrm{b1}}{,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{\le }{\mathrm{b2}}\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{a}{<}{\mathrm{b1}}{,}{\mathrm{c1}}{\le }{\mathrm{b1}}{,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{\le }{a}\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{\le }{a}{,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{a}{<}{\mathrm{b2}}{,}{\mathrm{c2}}{\le }{\mathrm{b2}}\right]\right]'$ (6)

Likewise, the conditions for the singular cases of AppellF2 can be seen either using the FunctionAdvisor or entering AppellF2:-Singularities(), so with no arguments.

For particular values of its parameters, AppellF2 is related to the hypergeometric function. These hypergeometric cases are returned automatically. For example, for ${c}_{1}={b}_{1}$,

 > $\left(\mathrm{%AppellF2}=\mathrm{AppellF2}\right)\left(a,\mathrm{b__1},\mathrm{b__2},\mathrm{b__1},\mathrm{c__2},\mathrm{z__1},\mathrm{z__2}\right)$
 ${\mathrm{%AppellF2}}{}\left({a}{,}\mathit{b__1}{,}\mathit{b__2}{,}\mathit{b__1}{,}\mathit{c__2}{,}\mathit{z__1}{,}\mathit{z__2}\right){=}{\left({1}{-}\mathrm{z__1}\right)}^{{-}{a}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\frac{\mathrm{z__2}}{{1}{-}\mathrm{z__1}}\right)$ (7)

To see all the hypergeometric cases, enter

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{AppellF2},\mathrm{hypergeom}\right)$
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\mathrm{z__2}\right){,}\mathrm{z__1}{=}{0}{\vee }\mathrm{b__1}{=}{0}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}\mathrm{z__1}\right){,}\mathrm{z__2}{=}{0}{\vee }\mathrm{b__2}{=}{0}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\frac{\mathrm{z__2}}{{1}{-}\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{{a}}}{,}\mathrm{c__1}{=}\mathrm{b__1}{\wedge }\mathrm{z__1}{\ne }{1}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}\frac{\mathrm{z__1}}{{1}{-}\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{{a}}}{,}\mathrm{c__2}{=}\mathrm{b__2}{\wedge }\mathrm{z__2}{\ne }{1}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}\frac{\mathrm{b__2}}{{2}}{+}\frac{\mathrm{b__1}}{{2}}{,}\frac{\mathrm{b__2}}{{2}}{+}\frac{\mathrm{b__1}}{{2}}{+}\frac{{1}}{{2}}{;}\frac{{1}}{{2}}{+}\mathrm{b__1}{,}\mathrm{b__2}{+}\frac{{1}}{{2}}{,}\mathrm{b__2}{+}\mathrm{b__1}{;}{\mathrm{z__1}}^{{2}}\right){,}\mathrm{c__1}{=}{2}{}\mathrm{b__1}{\wedge }\mathrm{c__2}{=}{2}{}\mathrm{b__2}{\wedge }\mathrm{z__2}{=}{-}\mathrm{z__1}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left(\mathrm{b__1}{,}\frac{{a}}{{2}}{,}\mathrm{c__1}{-}\mathrm{b__1}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{;}\mathrm{c__1}{,}\frac{\mathrm{c__1}}{{2}}{,}\frac{\mathrm{c__1}}{{2}}{+}\frac{{1}}{{2}}{;}{\mathrm{z__1}}^{{2}}\right){,}\mathrm{b__2}{=}\mathrm{b__1}{\wedge }\mathrm{c__2}{=}\mathrm{c__1}{\wedge }\mathrm{z__2}{=}{-}\mathrm{z__1}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}\frac{\mathrm{c__1}}{{2}}{+}\frac{{1}}{{2}}{-}\mathrm{b__1}{,}\frac{{1}}{{2}}{-}\frac{\mathrm{c__1}}{{2}}{+}\mathrm{b__1}{;}\frac{{1}}{{2}}{,}\frac{\mathrm{c__1}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{-}\frac{\mathrm{c__1}}{{2}}{;}{\mathrm{z__1}}^{{2}}\right){-}\frac{{a}{}\left({1}{-}\mathrm{c__1}\right){}\left(\mathrm{c__1}{-}{2}{}\mathrm{b__1}\right){}\mathrm{z__1}{}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{+}{1}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}{1}{-}\frac{\mathrm{c__1}}{{2}}{+}\mathrm{b__1}{,}{1}{+}\frac{\mathrm{c__1}}{{2}}{-}\mathrm{b__1}{;}\frac{{3}}{{2}}{,}{2}{-}\frac{\mathrm{c__1}}{{2}}{,}{1}{+}\frac{\mathrm{c__1}}{{2}}{;}{\mathrm{z__1}}^{{2}}\right)}{\mathrm{c__1}{}\left({2}{-}\mathrm{c__1}\right)}{,}\left(\mathrm{b__2}{=}{1}{+}\mathrm{b__1}{-}\mathrm{c__1}{\wedge }\mathrm{c__2}{=}{2}{-}\mathrm{c__1}{\wedge }\mathrm{z__2}{=}{-}\mathrm{z__1}{\wedge }\mathrm{c__1}{\ne }{2}\right){\vee }\left(\mathrm{b__1}{=}{1}{+}\mathrm{b__2}{-}\mathrm{c__2}{\wedge }\mathrm{c__1}{=}{2}{-}\mathrm{c__2}{\wedge }\mathrm{z__2}{=}{-}\mathrm{z__1}{\wedge }\mathrm{c__2}{\ne }{2}\right)\right]$ (8)

Other special values of AppellF2 can be seen using FunctionAdvisor(special_values, AppellF2).

By requesting the sum form of AppellF2, besides its double power series definition, we also see the particular form the series takes when one of the summations is performed and the result expressed in terms of 2F1 hypergeometric functions:

 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{AppellF2}\right)$
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{m}{+}{n}}{}{\left(\mathrm{b__1}\right)}_{{m}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left(\mathrm{c__1}\right)}_{{m}}{}{\left(\mathrm{c__2}\right)}_{{n}}{}{m}{!}{}{n}{!}}{,}\left|\mathrm{z__2}\right|{+}\left|\mathrm{z__1}\right|{<}{1}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__1}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left(\mathrm{c__1}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__2}\right|{+}\left|\mathrm{z__1}\right|{<}{1}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__2}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left(\mathrm{c__2}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__2}\right|{+}\left|\mathrm{z__1}\right|{<}{1}\right]$ (9)

As indicated in the formulas above, for AppellF2 (also for AppellF4), and unlike the case of AppellF1 and AppellF3, the domain of convergence with regards to the two variables ${z}_{1}$ and ${z}_{2}$ is entangled, i.e. it intrinsically depends on a combination of the two variables, so the hypergeometric coefficient in one variable in the single sum form does not extend the domain of convergence of the double sum but for particular cases, and from the formulas above one cannot conclude about the value of the function when one of ${z}_{1}$ or ${z}_{2}$ is equal to 1 unless the other one is exactly equal to 0.

AppellF2 admits identities analogous to Euler identities for the hypergeometric function. These Euler-type identities, as well as contiguity identities, are visible using the FunctionAdvisor with the option identities, or directly from the function. For example,

 >
 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{2}}{}\left({a}{,}\mathrm{c__1}{-}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\frac{\mathrm{z__1}}{\mathrm{z__1}{-}{1}}{,}\frac{\mathrm{z__2}}{{1}{-}\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{{a}}}$ (10)

Among other situations, this identity is useful when the sum of the absolute values of ${z}_{1}$ and ${z}_{2}$ is larger than 1 but the same sum constructed with the arguments in the same position of AppellF2 on the right-hand side is smaller than 1. On the other hand, unlike the case of the other three Appell functions, none of the two Euler type transformations or hypergeometric special cases of AppellF2 are of help to analytically extend to the whole complex plane the AppellF2 series when either ${z}_{1}=1$ or ${z}_{2}=1$.

A contiguity transformation for AppellF2

 >
 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{-}{a}{}{{F}}_{{2}}{}\left({a}{+}{1}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){+}\mathrm{b__1}{}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){+}\mathrm{b__2}{}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{+}{1}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{-}{a}{+}\mathrm{b__1}{+}\mathrm{b__2}}$ (11)

The contiguity transformations available in this way are

 > $\mathrm{indices}\left(\mathrm{AppellF2}:-\mathrm{Transformations}\left["Contiguity"\right]\right)$
 $\left[{1}\right]{,}\left[{2}\right]{,}\left[{3}\right]{,}\left[{4}\right]{,}\left[{5}\right]{,}\left[{6}\right]{,}\left[{7}\right]{,}\left[{8}\right]$ (12)

By using differential algebra techniques, the PDE system satisfied by AppellF2 can be transformed into an equivalent PDE system where one of the equations is a linear ODE in ${z}_{2}$ parametrized by ${z}_{1}$. In the case of AppellF2 this linear ODE is of fourth order and can be computed as follows

 >
 ${\mathrm{F2}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ (13)
 >
 $\frac{{{\partial }}^{{4}}}{{\partial }{\mathrm{z__2}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F2}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{-}\left(\left({2}{}{a}{+}{2}{}\mathrm{b__2}{-}\mathrm{c__1}{+}{8}\right){}{\mathrm{z__2}}^{{2}}{+}\left(\left(\mathrm{z__1}{-}{2}\right){}\mathrm{b__2}{+}\left(\mathrm{z__1}{-}{2}\right){}{a}{+}\left(\mathrm{z__1}{-}{2}\right){}\mathrm{c__2}{+}\left({-}\mathrm{b__1}{+}{5}\right){}\mathrm{z__1}{+}\mathrm{c__1}{-}{10}\right){}\mathrm{z__2}{-}{2}{}\left(\mathrm{z__1}{-}{1}\right){}\left(\mathrm{c__2}{+}{1}\right)\right){}\mathrm{z__2}{}\left(\frac{{{\partial }}^{{3}}}{{\partial }{\mathrm{z__2}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F2}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right){+}\left(\left({-}{\mathrm{b__2}}^{{2}}{+}\left({-}{4}{}{a}{+}{2}{}\mathrm{c__1}{-}{9}\right){}\mathrm{b__2}{-}{{a}}^{{2}}{+}\left(\mathrm{c__1}{-}{9}\right){}{a}{+}{4}{}\mathrm{c__1}{-}{14}\right){}{\mathrm{z__2}}^{{2}}{+}\left(\left(\left({-}\mathrm{z__1}{+}{2}\right){}{a}{+}\left({-}\mathrm{z__1}{+}{2}\right){}\mathrm{c__2}{+}\left(\mathrm{b__1}{-}{2}\right){}\mathrm{z__1}{-}\mathrm{c__1}{+}{4}\right){}\mathrm{b__2}{-}\left(\mathrm{z__1}{-}{2}\right){}\left(\mathrm{c__2}{+}{2}\right){}{a}{+}\left(\left(\mathrm{b__1}{-}{3}\right){}\mathrm{z__1}{-}\mathrm{c__1}{+}{6}\right){}\mathrm{c__2}{+}\left({2}{}\mathrm{b__1}{-}{4}\right){}\mathrm{z__1}{-}{2}{}\mathrm{c__1}{+}{8}\right){}\mathrm{z__2}{+}\mathrm{c__2}{}\left(\mathrm{z__1}{-}{1}\right){}\left(\mathrm{c__2}{+}{1}\right)\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__2}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F2}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right){-}\left(\left(\left(\left({2}{}{a}{-}\mathrm{c__1}{+}{2}\right){}\mathrm{b__2}{+}{2}{}\left({a}{+}{1}\right){}\left({a}{-}\mathrm{c__1}{+}{2}\right)\right){}\mathrm{z__2}{+}\mathrm{c__2}{}\left(\left(\mathrm{z__1}{-}{2}\right){}{a}{+}\left({-}\mathrm{b__1}{+}{1}\right){}\mathrm{z__1}{+}\mathrm{c__1}{-}{2}\right)\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F2}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right){+}{\mathrm{F2}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}{}\mathrm{b__2}{}\left({a}{-}\mathrm{c__1}{+}{1}\right)\right){}\left(\mathrm{b__2}{+}{1}\right)}{{\mathrm{z__2}}^{{2}}{}\left(\mathrm{z__2}{-}{1}\right){}\left(\mathrm{z__2}{+}\mathrm{z__1}{-}{1}\right)}$ (14)

This linear ODE has four regular singularities, one of which is depends on ${z}_{1}$

 > $\mathrm{DEtools}\left[\mathrm{singularities}\right]\left(\mathrm{subs}\left(\mathrm{F2}\left(\mathrm{z__1},\mathrm{z__2}\right)=\mathrm{F2}\left(\mathrm{z__2}\right),\right)\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}{,}{\mathrm{\infty }}{,}{1}{-}\mathrm{z__1}\right\}{,}{\mathrm{irregular}}{=}{\varnothing }$ (15)

You can also see a general presentation of AppellF2, organized into sections and including plots, using the FunctionAdvisor

 > $\mathrm{FunctionAdvisor}\left(\mathrm{AppellF2}\right)$

AppellF2

describe

 ${\mathrm{AppellF2}}{=}{\mathrm{Appell 2-variable hypergeometric function F2}}$

definition

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left(\mathrm{c__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{c__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}$ $\left|\mathrm{z__1}\right|{+}\left|\mathrm{z__2}\right|{<}{1}$

classify function

 ${\mathrm{Appell}}$

symmetries

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}\mathrm{c__2}{,}\mathrm{c__1}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$

plot

singularities

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left(\mathrm{c__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right)\right){\vee }\left(\mathrm{c__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{a}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__1}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{<}\mathrm{c__1}{\wedge }\mathrm{b__1}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right)\right){\vee }\left(\mathrm{c__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{a}{<}\mathrm{c__2}\right){\vee }\left(\mathrm{c__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__2}{<}\mathrm{c__2}\right){\vee }\left(\mathrm{c__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{<}\mathrm{c__2}{\wedge }\mathrm{b__2}{<}\mathrm{c__2}\right)$

branch points

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left({a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{z__1}{\in }\left[{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]\right){\vee }\left({a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{z__2}{\in }\left[{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]\right)$

branch cuts

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left({a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{1}{<}\mathrm{z__1}\right){\vee }\left({a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{1}{<}\mathrm{z__2}\right)$

special values

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ $\mathrm{z__1}{=}{0}{\wedge }\mathrm{z__2}{=}{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ ${a}{=}{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ $\mathrm{b__1}{=}{0}{\wedge }\mathrm{b__2}{=}{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\mathrm{z__2}\right)$ $\mathrm{z__1}{=}{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}\mathrm{z__1}\right)$ $\mathrm{z__2}{=}{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\mathrm{z__2}\right)$ $\mathrm{b__1}{=}{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}\mathrm{z__1}\right)$ $\mathrm{b__2}{=}{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{-}\mathrm{z__1}\right)}^{{-}{a}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\frac{\mathrm{z__2}}{{1}{-}\mathrm{z__1}}\right)$ $\mathrm{c__1}{=}\mathrm{b__1}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{-}\mathrm{z__2}\right)}^{{-}{a}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}\frac{\mathrm{z__1}}{{1}{-}\mathrm{z__2}}\right)$ $\mathrm{c__2}{=}\mathrm{b__2}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}\frac{\mathrm{b__2}}{{2}}{+}\frac{\mathrm{b__1}}{{2}}{,}\frac{\mathrm{b__2}}{{2}}{+}\frac{\mathrm{b__1}}{{2}}{+}\frac{{1}}{{2}}{;}\mathrm{b__2}{+}\mathrm{b__1}{,}\frac{{1}}{{2}}{+}\mathrm{b__1}{,}\mathrm{b__2}{+}\frac{{1}}{{2}}{;}{\mathrm{z__1}}^{{2}}\right)$ $\mathrm{c__1}{=}{2}{}\mathrm{b__1}{\wedge }\mathrm{c__2}{=}{2}{}\mathrm{b__2}{\wedge }\mathrm{z__2}{=}{-}\mathrm{z__1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left(\mathrm{b__1}{,}\frac{{a}}{{2}}{,}\mathrm{c__1}{-}\mathrm{b__1}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{;}\mathrm{c__1}{,}\frac{\mathrm{c__1}}{{2}}{,}\frac{\mathrm{c__1}}{{2}}{+}\frac{{1}}{{2}}{;}{\mathrm{z__1}}^{{2}}\right)$ $\mathrm{b__2}{=}\mathrm{b__1}{\wedge }\mathrm{c__2}{=}\mathrm{c__1}{\wedge }\mathrm{z__2}{=}{-}\mathrm{z__1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}\frac{\mathrm{c__1}}{{2}}{+}\frac{{1}}{{2}}{-}\mathrm{b__1}{,}\frac{{1}}{{2}}{-}\frac{\mathrm{c__1}}{{2}}{+}\mathrm{b__1}{;}\frac{{1}}{{2}}{,}\frac{\mathrm{c__1}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{-}\frac{\mathrm{c__1}}{{2}}{;}{\mathrm{z__1}}^{{2}}\right){-}\frac{{a}{}\left({1}{-}\mathrm{c__1}\right){}\left(\mathrm{c__1}{-}{2}{}\mathrm{b__1}\right){}\mathrm{z__1}{}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{+}{1}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}{1}{-}\frac{\mathrm{c__1}}{{2}}{+}\mathrm{b__1}{,}{1}{+}\frac{\mathrm{c__1}}{{2}}{-}\mathrm{b__1}{;}\frac{{3}}{{2}}{,}{2}{-}\frac{\mathrm{c__1}}{{2}}{,}{1}{+}\frac{\mathrm{c__1}}{{2}}{;}{\mathrm{z__1}}^{{2}}\right)}{\mathrm{c__1}{}\left({2}{-}\mathrm{c__1}\right)}$ $\mathrm{b__2}{=}{1}{+}\mathrm{b__1}{-}\mathrm{c__1}{\wedge }\mathrm{c__2}{=}{2}{-}\mathrm{c__1}{\wedge }\mathrm{z__2}{=}{-}\mathrm{z__1}{\wedge }\mathrm{c__1}{\ne }{2}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}\frac{\mathrm{c__2}}{{2}}{+}\frac{{1}}{{2}}{-}\mathrm{b__2}{,}\frac{{1}}{{2}}{-}\frac{\mathrm{c__2}}{{2}}{+}\mathrm{b__2}{;}\frac{{1}}{{2}}{,}\frac{\mathrm{c__2}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{3}}{{2}}{-}\frac{\mathrm{c__2}}{{2}}{;}{\mathrm{z__2}}^{{2}}\right){-}\frac{{a}{}\left({1}{-}\mathrm{c__2}\right){}\left(\mathrm{c__2}{-}{2}{}\mathrm{b__2}\right){}\mathrm{z__2}{}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{+}{1}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}{1}{-}\frac{\mathrm{c__2}}{{2}}{+}\mathrm{b__2}{,}{1}{+}\frac{\mathrm{c__2}}{{2}}{-}\mathrm{b__2}{;}\frac{{3}}{{2}}{,}{2}{-}\frac{\mathrm{c__2}}{{2}}{,}{1}{+}\frac{\mathrm{c__2}}{{2}}{;}{\mathrm{z__2}}^{{2}}\right)}{\mathrm{c__2}{}\left({2}{-}\mathrm{c__2}\right)}$ $\mathrm{b__1}{=}{1}{+}\mathrm{b__2}{-}\mathrm{c__2}{\wedge }\mathrm{c__1}{=}{2}{-}\mathrm{c__2}{\wedge }\mathrm{z__2}{=}{-}\mathrm{z__1}{\wedge }\mathrm{c__2}{\ne }{2}$

identities

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{2}}{}\left({a}{,}\mathrm{c__1}{-}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\frac{\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}}{,}\frac{\mathrm{z__2}}{{1}{-}\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{{a}}}$ $\mathrm{z__1}{\ne }{1}{\wedge }\left({a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\vee }\left(\mathrm{b__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\vee }{¬}\left({1}{<}\mathrm{z__1}{\wedge }{1}{<}\mathrm{z__2}\right)\right)$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{2}}{}\left({a}{,}\mathrm{c__1}{-}\mathrm{b__1}{,}\mathrm{c__2}{-}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\frac{\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}}{,}\frac{\mathrm{z__2}}{{-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__1}{-}\mathrm{z__2}\right)}^{{a}}}$ $\mathrm{z__2}{\ne }{1}{-}\mathrm{z__1}{\wedge }\left({a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\vee }\left(\mathrm{b__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\vee }{¬}\left({1}{<}\mathrm{z__1}{\vee }{1}{<}\mathrm{z__2}\right)\right)$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{-}{a}{}{{F}}_{{2}}{}\left({a}{+}{1}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){+}\mathrm{b__1}{}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){+}\mathrm{b__2}{}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{+}{1}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\mathrm{b__2}{+}\mathrm{b__1}{-}{a}}$ $\mathrm{z__2}{\ne }{1}{\wedge }\mathrm{z__1}{\ne }{1}{\wedge }{a}{\ne }\mathrm{b__2}{+}\mathrm{b__1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{-}\frac{\mathrm{b__1}{}\mathrm{z__1}{}\left({\sum }_{{k}{=}{0}}^{{n}{-}{1}}{}{{F}}_{{2}}{}\left({a}{+}{k}{+}{1}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}\mathrm{c__1}{+}{1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{c__1}}{-}\frac{\mathrm{b__2}{}\mathrm{z__2}{}\left({\sum }_{{k}{=}{0}}^{{n}{-}{1}}{}{{F}}_{{2}}{}\left({a}{+}{k}{+}{1}{,}\mathrm{b__1}{,}\mathrm{b__2}{+}{1}{,}\mathrm{c__1}{,}\mathrm{c__2}{+}{1}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{c__2}}{+}{{F}}_{{2}}{}\left({a}{+}{n}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\mathrm{c__1}{\ne }{0}{\wedge }\mathrm{c__2}{\ne }{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{+}{n}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}\frac{{a}{}\mathrm{z__1}{}\left({\sum }_{{k}{=}{1}}^{{n}}{}{{F}}_{{2}}{}\left({a}{+}{1}{,}\mathrm{b__1}{+}{k}{,}\mathrm{b__2}{,}\mathrm{c__1}{+}{1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{c__1}}$ $\mathrm{c__1}{\ne }{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{-}{n}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}{a}{}\mathrm{b__1}{}\mathrm{z__1}{}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{{{F}}_{{2}}{}\left({a}{+}{1}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}\mathrm{c__1}{-}{k}{+}{2}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\left(\mathrm{c__1}{-}{k}\right){}\left(\mathrm{c__1}{-}{k}{+}{1}\right)}\right)$ $\mathrm{c__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{posint}}{'}\right]\right)\right){\vee }{n}{<}\mathrm{c__1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left({-1}\right)}^{{n}}{}{\left({a}\right)}_{{n}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{\mathrm{z__2}}^{{n}}{}{{F}}_{{2}}{}\left({a}{+}{n}{,}\mathrm{b__1}{,}{n}{+}\mathrm{b__2}{,}\mathrm{c__1}{,}{n}{+}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left(\mathrm{c__2}\right)}_{{n}}{}{\left(\mathrm{c__2}{-}{1}\right)}_{{n}}}{-}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{n}}{{k}}\right){}{\left({1}{-}\mathrm{c__2}\right)}_{{k}}{}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{-}{k}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left({2}{-}\mathrm{c__2}{-}{n}\right)}_{{k}}}\right)$ $\mathrm{c__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{integer}}{'}\right]\right)\right){\vee }{n}{<}\mathrm{c__2}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{b__1}{}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{1}{+}\mathrm{b__1}{-}\mathrm{c__1}}{-}\frac{{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{-}{1}{+}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}\left({-}{1}{+}\mathrm{c__1}\right)}{{1}{+}\mathrm{b__1}{-}\mathrm{c__1}}$ $\mathrm{c__1}{\ne }{1}{\wedge }{1}{+}\mathrm{b__1}{-}\mathrm{c__1}{\ne }{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}\mathrm{c__1}{+}{1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}\mathrm{b__1}}{\mathrm{c__1}}{-}\frac{\left(\mathrm{b__1}{-}\mathrm{c__1}\right){}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{+}{1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\mathrm{c__1}}$ $\mathrm{c__1}{\ne }{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{+}{1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){+}\frac{{a}{}\mathrm{z__1}{}{{F}}_{{2}}{}\left({a}{+}{1}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{+}{1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\mathrm{c__1}}{+}\frac{{{F}}_{{2}}{}\left({a}{+}{1}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{+}{2}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}{}\left(\mathrm{b__1}{-}\mathrm{c__1}{-}{1}\right){}\mathrm{z__1}}{\mathrm{c__1}{}\left(\mathrm{c__1}{+}{1}\right)}$ $\mathrm{c__1}{\ne }{0}{\wedge }\mathrm{c__1}{\ne }{-1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{b__1}{,}{-}\mathrm{b__2}{+}{a}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{z__1}{,}\frac{\mathrm{z__1}}{{1}{-}\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{b__2}}}$ $\mathrm{c__2}{=}{a}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{b__2}{,}{a}{-}\mathrm{b__1}{,}\mathrm{b__1}{,}\mathrm{c__2}{,}\mathrm{z__2}{,}\frac{\mathrm{z__2}}{{1}{-}\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__1}}}$ $\mathrm{c__1}{=}{a}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{{-}{1}{+}\mathrm{z__2}}{{-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}}\right)}^{\mathrm{b__2}}{}{{F}}_{{3}}{}\left(\mathrm{b__1}{,}\mathrm{b__2}{,}{-}\mathrm{b__2}{+}{a}{,}\mathrm{c__1}{-}\mathrm{b__1}{,}\mathrm{c__1}{,}\mathrm{z__1}{,}\frac{\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{b__2}}}$ $\mathrm{c__2}{=}{a}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{{-}{1}{+}\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}}\right)}^{\mathrm{b__1}}{}{{F}}_{{3}}{}\left(\mathrm{b__2}{,}\mathrm{b__1}{,}{a}{-}\mathrm{b__1}{,}\mathrm{c__2}{-}\mathrm{b__2}{,}\mathrm{c__2}{,}\mathrm{z__2}{,}\frac{\mathrm{z__2}}{{-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__1}}}$ $\mathrm{c__1}{=}{a}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{4}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{}\left({1}{-}\mathrm{z__2}\right){,}\mathrm{z__2}{}\left({1}{-}\mathrm{z__1}\right)\right)$ ${a}{=}\mathrm{c__1}{+}\mathrm{c__2}{-}{1}{\wedge }\mathrm{b__1}{=}\mathrm{b__2}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{4}}{}\left(\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}\mathrm{b__1}{,}\mathrm{b__2}{+}\frac{{1}}{{2}}{,}\frac{{\mathrm{z__1}}^{{2}}}{{\left({-}{2}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}{,}\frac{{\mathrm{z__2}}^{{2}}}{{\left({-}{2}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}{{\left({1}{-}\frac{\mathrm{z__1}}{{2}}{-}\frac{\mathrm{z__2}}{{2}}\right)}^{{a}}}$ $\mathrm{c__1}{=}{2}{}\mathrm{b__1}{\wedge }\mathrm{c__2}{=}{2}{}\mathrm{b__2}{\wedge }{1}{-}\frac{\mathrm{z__1}}{{2}}{-}\frac{\mathrm{z__2}}{{2}}{\ne }{0}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{+}\sqrt{\frac{\left({-}{4}{}\mathrm{z__2}{+}{8}\right){}\sqrt{{1}{-}\mathrm{z__2}}{+}{\mathrm{z__2}}^{{2}}{-}{8}{}\mathrm{z__2}{+}{8}}{{\mathrm{z__2}}^{{2}}}}\right)}^{{2}{}{a}}{}{{F}}_{{4}}{}\left({a}{,}\mathrm{b__1}{,}{a}{-}\mathrm{b__1}{+}{1}{,}\mathrm{c__1}{,}\frac{\left({-}{4}{}\mathrm{z__2}{+}{8}\right){}\sqrt{{1}{-}\mathrm{z__2}}{+}{\mathrm{z__2}}^{{2}}{-}{8}{}\mathrm{z__2}{+}{8}}{{\mathrm{z__2}}^{{2}}}{,}\frac{{2}{}\mathrm{z__1}{}\left(\sqrt{\frac{\left({-}{4}{}\mathrm{z__2}{+}{8}\right){}\sqrt{{1}{-}\mathrm{z__2}}{+}{\mathrm{z__2}}^{{2}}{-}{8}{}\mathrm{z__2}{+}{8}}{{\mathrm{z__2}}^{{2}}}}{}{\mathrm{z__2}}^{{2}}{+}\left(\mathrm{z__2}{-}{2}\right){}\left(\mathrm{z__2}{-}{2}{}\sqrt{{1}{-}\mathrm{z__2}}{-}{2}\right)\right)}{{\mathrm{z__2}}^{{2}}}\right)$ $\mathrm{b__2}{=}{a}{-}\mathrm{b__1}{+}\frac{{1}}{{2}}{\wedge }\mathrm{c__2}{=}{2}{}\mathrm{b__2}{\wedge }{\left({\mathrm{\Re }}{}\left(\mathrm{z__2}\right){-}{1}\right)}^{{2}}{+}{{\mathrm{\Im }}{}\left(\mathrm{z__2}\right)}^{{2}}{\le }{1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{+}\sqrt{\frac{\left({-}{4}{}\mathrm{z__1}{+}{8}\right){}\sqrt{{1}{-}\mathrm{z__1}}{+}{\mathrm{z__1}}^{{2}}{-}{8}{}\mathrm{z__1}{+}{8}}{{\mathrm{z__1}}^{{2}}}}\right)}^{{2}{}{a}}{}{{F}}_{{4}}{}\left({a}{,}\mathrm{b__2}{,}{a}{-}\mathrm{b__2}{+}{1}{,}\mathrm{c__2}{,}\frac{\left({-}{4}{}\mathrm{z__1}{+}{8}\right){}\sqrt{{1}{-}\mathrm{z__1}}{+}{\mathrm{z__1}}^{{2}}{-}{8}{}\mathrm{z__1}{+}{8}}{{\mathrm{z__1}}^{{2}}}{,}\frac{{2}{}\mathrm{z__2}{}\left(\sqrt{\frac{\left({-}{4}{}\mathrm{z__1}{+}{8}\right){}\sqrt{{1}{-}\mathrm{z__1}}{+}{\mathrm{z__1}}^{{2}}{-}{8}{}\mathrm{z__1}{+}{8}}{{\mathrm{z__1}}^{{2}}}}{}{\mathrm{z__1}}^{{2}}{+}\left(\mathrm{z__1}{-}{2}\right){}\left(\mathrm{z__1}{-}{2}{}\sqrt{{1}{-}\mathrm{z__1}}{-}{2}\right)\right)}{{\mathrm{z__1}}^{{2}}}\right)$ $\mathrm{b__1}{=}{a}{-}\mathrm{b__2}{+}\frac{{1}}{{2}}{\wedge }\mathrm{c__1}{=}{2}{}\mathrm{b__1}{\wedge }{\left({\mathrm{\Re }}{}\left(\mathrm{z__1}\right){-}{1}\right)}^{{2}}{+}{{\mathrm{\Im }}{}\left(\mathrm{z__1}\right)}^{{2}}{\le }{1}$

sum form

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{m}{+}{n}}{}{\left(\mathrm{b__1}\right)}_{{m}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left(\mathrm{c__1}\right)}_{{m}}{}{\left(\mathrm{c__2}\right)}_{{n}}{}{m}{!}{}{n}{!}}$ $\left|\mathrm{z__2}\right|{+}\left|\mathrm{z__1}\right|{<}{1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__1}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left(\mathrm{c__1}\right)}_{{k}}{}{k}{!}}$ $\left|\mathrm{z__2}\right|{+}\left|\mathrm{z__1}\right|{<}{1}$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__2}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left(\mathrm{c__2}\right)}_{{k}}{}{k}{!}}$ $\left|\mathrm{z__2}\right|{+}\left|\mathrm{z__1}\right|{<}{1}$

series

 ${\mathrm{series}}{}\left({{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\mathrm{z__1}{,}{4}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\mathrm{z__2}\right){+}\frac{{a}{}\mathrm{b__1}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{b__2}{,}{a}{+}{1}{;}\mathrm{c__2}{;}\mathrm{z__2}\right)}{\mathrm{c__1}}{}\mathrm{z__1}{+}\frac{{1}}{{2}}{}\frac{{a}{}\mathrm{b__1}{}\left({a}{+}{1}\right){}\left(\mathrm{b__1}{+}{1}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{b__2}{,}{a}{+}{2}{;}\mathrm{c__2}{;}\mathrm{z__2}\right)}{\mathrm{c__1}{}\left(\mathrm{c__1}{+}{1}\right)}{}{\mathrm{z__1}}^{{2}}{+}\frac{{1}}{{6}}{}\frac{{a}{}\mathrm{b__1}{}\left({a}{+}{1}\right){}\left(\mathrm{b__1}{+}{1}\right){}\left({a}{+}{2}\right){}\left(\mathrm{b__1}{+}{2}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{b__2}{,}{a}{+}{3}{;}\mathrm{c__2}{;}\mathrm{z__2}\right)}{\mathrm{c__1}{}\left(\mathrm{c__1}{+}{1}\right){}\left(\mathrm{c__1}{+}{2}\right)}{}{\mathrm{z__1}}^{{3}}{+}{O}{}\left({\mathrm{z__1}}^{{4}}\right)$

 ${\mathrm{series}}{}\left({{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\mathrm{z__2}{,}{4}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}\mathrm{z__1}\right){+}\frac{{a}{}\mathrm{b__2}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{b__1}{,}{a}{+}{1}{;}\mathrm{c__1}{;}\mathrm{z__1}\right)}{\mathrm{c__2}}{}\mathrm{z__2}{+}\frac{{1}}{{2}}{}\frac{{a}{}\mathrm{b__2}{}\left({a}{+}{1}\right){}\left(\mathrm{b__2}{+}{1}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{b__1}{,}{a}{+}{2}{;}\mathrm{c__1}{;}\mathrm{z__1}\right)}{\mathrm{c__2}{}\left(\mathrm{c__2}{+}{1}\right)}{}{\mathrm{z__2}}^{{2}}{+}\frac{{1}}{{6}}{}\frac{{a}{}\mathrm{b__2}{}\left({a}{+}{1}\right){}\left(\mathrm{b__2}{+}{1}\right){}\left({a}{+}{2}\right){}\left(\mathrm{b__2}{+}{2}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{b__1}{,}{a}{+}{3}{;}\mathrm{c__1}{;}\mathrm{z__1}\right)}{\mathrm{c__2}{}\left(\mathrm{c__2}{+}{1}\right){}\left(\mathrm{c__2}{+}{2}\right)}{}{\mathrm{z__2}}^{{3}}{+}{O}{}\left({\mathrm{z__2}}^{{4}}\right)$

integral form

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__1}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{{u}}^{\mathrm{b__1}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}{-}\frac{\mathrm{z__2}}{{u}{}\mathrm{z__1}{-}{1}}\right)}{{\left({1}{-}{u}\right)}^{{-}\mathrm{c__1}{+}\mathrm{b__1}{+}{1}}{}{\left({-}{u}{}\mathrm{z__1}{+}{1}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__1}{-}\mathrm{b__1}\right)}$ $\mathrm{z__1}{\ne }{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__1}{+}\mathrm{b__1}\right)$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{{u}}^{\mathrm{b__2}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}{-}\frac{\mathrm{z__1}}{{u}{}\mathrm{z__2}{-}{1}}\right)}{{\left({1}{-}{u}\right)}^{{1}{+}\mathrm{b__2}{-}\mathrm{c__2}}{}{\left({-}{u}{}\mathrm{z__2}{+}{1}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}\mathrm{b__2}\right)}$ $\mathrm{z__2}{\ne }{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__2}{+}\mathrm{b__2}\right)$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}\left({{\int }}_{{0}}^{{1}}{{\int }}_{{0}}^{{1}}\frac{{{u}}^{\mathrm{b__1}{-}{1}}{}{{v}}^{\mathrm{b__2}{-}{1}}}{{\left({1}{-}{u}\right)}^{{-}\mathrm{c__1}{+}\mathrm{b__1}{+}{1}}{}{\left({1}{-}{v}\right)}^{{1}{+}\mathrm{b__2}{-}\mathrm{c__2}}{}{\left({-}{u}{}\mathrm{z__1}{-}{v}{}\mathrm{z__2}{+}{1}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{v}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__1}{-}\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}\mathrm{b__2}\right)}$ ${0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__1}{+}\mathrm{b__1}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__2}{+}\mathrm{b__2}\right)$

 ${{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{\int }}_{{0}}^{{\mathrm{\infty }}}\frac{{{u}}^{{a}{-}{1}}{}{}_{{1}}{F}_{{1}}{}\left(\mathrm{b__1}{;}\mathrm{c__1}{;}{u}{}\mathrm{z__1}\right){}{}_{{1}}{F}_{{1}}{}\left(\mathrm{b__2}{;}\mathrm{c__2}{;}{u}{}\mathrm{z__2}\right)}{{{ⅇ}}^{{u}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}}{{\mathrm{\Gamma }}{}\left({a}\right)}$ ${\mathrm{\Re }}{}\left(\mathrm{z__1}{+}\mathrm{z__2}\right){<}{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left({a}\right)$

differentiation rule

 $\frac{{\partial }}{{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{a}{}\mathrm{b__1}{}{{F}}_{{2}}{}\left({a}{+}{1}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}\mathrm{c__1}{+}{1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\mathrm{c__1}}$

 $\frac{{{\partial }}^{{n}}}{{\partial }{\mathrm{z__1}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left({a}\right)}_{{n}}{}{\left(\mathrm{b__1}\right)}_{{n}}{}{{F}}_{{2}}{}\left({n}{+}{a}{,}{n}{+}\mathrm{b__1}{,}\mathrm{b__2}{,}{n}{+}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left(\mathrm{c__1}\right)}_{{n}}}$

 $\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{a}{}\mathrm{b__2}{}{{F}}_{{2}}{}\left({a}{+}{1}{,}\mathrm{b__1}{,}\mathrm{b__2}{+}{1}{,}\mathrm{c__1}{,}\mathrm{c__2}{+}{1}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\mathrm{c__2}}$

 $\frac{{{\partial }}^{{n}}}{{\partial }{\mathrm{z__2}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left({a}\right)}_{{n}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{{F}}_{{2}}{}\left({n}{+}{a}{,}\mathrm{b__1}{,}{n}{+}\mathrm{b__2}{,}\mathrm{c__1}{,}{n}{+}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left(\mathrm{c__2}\right)}_{{n}}}$

DE

${f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$

 $\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__1}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{-}\frac{\mathrm{z__2}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }\mathrm{z__2}{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{-}{1}}{+}\frac{\left(\left({-}{a}{-}\mathrm{b__1}{-}{1}\right){}\mathrm{z__1}{+}\mathrm{c__1}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{-}\frac{\mathrm{b__1}{}\mathrm{z__2}{}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{-}\frac{{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}{}\mathrm{b__1}}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}$ $\frac{{{\partial }}^{{2}}}{{\partial }\mathrm{z__2}{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\left({1}{-}\mathrm{z__2}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__2}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}}{-}\frac{\mathrm{b__2}{}\left(\frac{{\partial }}{{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__2}}{+}\frac{\left(\left({-}{a}{-}\mathrm{b__2}{-}{1}\right){}\mathrm{z__2}{+}\mathrm{c__2}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\mathrm{z__2}}{-}\frac{{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}{}\mathrm{b__2}}{\mathrm{z__1}{}\mathrm{z__2}}$

References

 [1] Appell, P.; Kampe de Feriet, J. Fonctions Hypergeometriques et hyperspheriques. Gauthier-Villars, 1926.
 [2] Srivastava, H. M.; Karlsson, P. W. Multiple Gaussian Hypergeometric Series. Ellis Horwood, 1985.

Compatibility

 • The AppellF2 command was introduced in Maple 2017.