AppellF1 - Maple Help

AppellF1

The AppellF1 function

 Calling Sequence AppellF1($a,{b}_{1},{b}_{2},c,{z}_{1},{z}_{2}$)

Parameters

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

Description

 • As is the case of all the four multi-parameter Appell functions, AppellF1, 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, AppellF1 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 AppellF1 series and the corresponding domain of convergence can be seen through the FunctionAdvisor
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left({c}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]$ (1)
 A distinction is made between the AppellF1 doubly hypergeometric series, with the restricted domain of convergence shown above, and the AppellF1 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 AppellF1 variables subscripted with the numbers 1 and 2, the function remains the same; hence
 > $\mathrm{FunctionAdvisor}\left(\mathrm{symmetries},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right]$ (2)
 From the series' definition, AppellF1 is singular (division by zero) when the $c$ parameter entering the pochhammer function in the denominator of the series is a non-positive integer because the pochhammer function will be equal to zero when the summation index of the series is bigger than the absolute value of $c$.
 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 AppellF1 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, AppellF1 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 (parameter $c$) in the pochhammer 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 any of the following conditions hold
 > $\mathrm{FunctionAdvisor}\left(\mathrm{singularities},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{a}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{a}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right)\right]$ (3)
 The AppellF1 series is analytically extended to the AppellF1 function defined over the whole complex plane using identities and mainly by integral representations in terms of Eulerian integrals:
 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral_form},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{\left({1}{-}{u}\right)}^{{-}{1}{+}\mathrm{b__1}}{}{\left({-}{u}{}\mathrm{z__1}{+}{1}\right)}^{{-}{c}{+}{a}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}{c}{-}\mathrm{b__1}{;}\mathrm{z__2}{}{u}\right)}{{{u}}^{{-}{c}{+}\mathrm{b__1}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left({c}{-}\mathrm{b__1}\right){}{\left({1}{-}\mathrm{z__1}\right)}^{{-}{c}{+}{a}{+}\mathrm{b__1}}}{,}{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}{c}{+}\mathrm{b__1}\right)\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{\left({1}{-}{u}\right)}^{\mathrm{b__2}{-}{1}}{}{\left({-}\mathrm{z__2}{}{u}{+}{1}\right)}^{{-}{c}{+}{a}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}{c}{-}\mathrm{b__2}{;}{u}{}\mathrm{z__1}\right)}{{{u}}^{{-}{c}{+}\mathrm{b__2}{+}{1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left({c}{-}\mathrm{b__2}\right){}{\left({1}{-}\mathrm{z__2}\right)}^{{-}{c}{+}{a}{+}\mathrm{b__2}}}{,}{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}{c}{+}\mathrm{b__2}\right)\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{{u}}^{{a}{-}{1}}}{{\left({1}{-}{u}\right)}^{{-}{c}{+}{a}{+}{1}}{}{\left({-}{u}{}\mathrm{z__1}{+}{1}\right)}^{\mathrm{b__1}}{}{\left({-}\mathrm{z__2}{}{u}{+}{1}\right)}^{\mathrm{b__2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left({a}\right){}{\mathrm{\Gamma }}{}\left({c}{-}{a}\right)}{,}{0}{<}{\mathrm{\Re }}{}\left({a}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}{c}{+}{a}\right)\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}{{\int }}_{{0}}^{{1}{-}{v}}\frac{{{u}}^{{-}{1}{+}\mathrm{b__1}}{}{{v}}^{\mathrm{b__2}{-}{1}}}{{\left({1}{-}{u}{-}{v}\right)}^{{-}{c}{+}\mathrm{b__1}{+}\mathrm{b__2}{+}{1}}{}{\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({c}{-}\mathrm{b__1}{-}\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\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({-}{c}{+}\mathrm{b__1}{+}\mathrm{b__2}\right)\right]$ (4)
 These integral representations are also the starting point for the derivation of many of the identities known for AppellF1.
 AppellF1 also satisfies a linear system of partial differential equations of second order
 > $\mathrm{FunctionAdvisor}\left(\mathrm{DE},\mathrm{AppellF1}\right)$
 $\left[{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\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}{,}{c}{,}\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}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}}{+}\frac{\left(\left({-}{a}{-}\mathrm{b__1}{-}{1}\right){}\mathrm{z__1}{+}{c}\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}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__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}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__1}\right)}{-}\frac{{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}{}\mathrm{b__1}}{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__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}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{-}\frac{\mathrm{z__2}{}\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}{,}{c}{,}\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}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__2}{-}{1}}{+}\frac{\left(\left({-}{a}{-}\mathrm{b__2}{-}{1}\right){}\mathrm{z__2}{+}{c}\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}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__2}{-}{1}\right)}{-}\frac{{f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}{}\mathrm{b__2}}{\mathrm{z__1}{}\left(\mathrm{z__2}{-}{1}\right)}\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 AppellF1. For example, the six polynomial cases of AppellF1 are

 > $\mathrm{AppellF1}:-\mathrm{SpecialValues}:-\mathrm{Polynomial}\left(\right)$
 ${6}{,}\left({a}{,}{\mathrm{b1}}{,}{\mathrm{b2}}{,}{c}{,}{\mathrm{z1}}{,}{\mathrm{z2}}\right){↦}'\left[\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{\le }{a}\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{\le }{\mathrm{b1}}{+}{\mathrm{b2}}\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]\right]'$ (6)

Likewise, the conditions for the singular cases of AppellF1 can be seen either using the FunctionAdvisor or entering AppellF1:-Singularities(), so with no arguments.

For particular values of its parameters, AppellF1 is related to the hypergeometric and elliptic functions. These hypergeometric cases are returned automatically. For example, for ${z}_{1}=1$,

 > $\left(\mathrm{%AppellF1}=\mathrm{AppellF1}\right)\left(a,\mathrm{b__1},\mathrm{b__2},c,1,\mathrm{z__2}\right)$
 ${\mathrm{%AppellF1}}{}\left({a}{,}\mathit{b__1}{,}\mathit{b__2}{,}{c}{,}{1}{,}\mathit{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}{c}{;}{1}\right){}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}{c}{-}\mathrm{b__1}{;}\mathrm{z__2}\right)$ (7)

This formula analytically extends to the whole complex plane the AppellF1 series when any of ${z}_{1}=1$ or ${z}_{2}=1$ (the latter using the symmetry of AppellF1 - see the beginning of the Description section).

To see all the hypergeometric cases, enter

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{AppellF1},\mathrm{hypergeom}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}{c}{;}\mathrm{z__2}\right){,}\mathrm{z__1}{=}{0}{\vee }\mathrm{b__1}{=}{0}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}{c}{;}\mathrm{z__1}\right){,}\mathrm{z__2}{=}{0}{\vee }\mathrm{b__2}{=}{0}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}{c}{;}{1}\right){}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}{c}{-}\mathrm{b__1}{;}\mathrm{z__2}\right){,}\mathrm{z__1}{=}{1}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}{c}{;}{1}\right){}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}{c}{-}\mathrm{b__2}{;}\mathrm{z__1}\right){,}\mathrm{z__2}{=}{1}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{+}\mathrm{b__2}{;}{c}{;}\mathrm{z__1}\right){,}\mathrm{z__1}{=}\mathrm{z__2}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{3}}{F}_{{2}}{}\left(\mathrm{b__1}{,}\frac{{a}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{;}\frac{{c}}{{2}}{,}\frac{{c}}{{2}}{+}\frac{{1}}{{2}}{;}{\mathrm{z__1}}^{{2}}\right){,}\mathrm{z__1}{=}{-}\mathrm{z__2}{\wedge }\mathrm{b__1}{=}\mathrm{b__2}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{1}}{F}_{{0}}{}\left(\mathrm{b__1}{;}{;}\mathrm{z__1}\right){}{}_{{1}}{F}_{{0}}{}\left(\mathrm{b__2}{;}{;}\mathrm{z__2}\right){,}{c}{=}{a}{\wedge }{a}{\ne }{0}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{z__1}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}{a}{;}{c}{;}\mathrm{z__1}\right){-}\mathrm{z__2}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}{a}{;}{c}{;}\mathrm{z__2}\right)}{{-}\mathrm{z__2}{+}\mathrm{z__1}}{,}\mathrm{b__1}{=}{1}{\wedge }\mathrm{b__2}{=}{1}{\wedge }\mathrm{z__1}{\ne }\mathrm{z__2}\right]$ (8)

Other special values of AppellF1 can be seen using FunctionAdvisor(special_values, AppellF1).

By requesting the sum form of AppellF1, 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:

 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{m}{+}{n}}{}{\left(\mathrm{b__1}\right)}_{{m}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left({c}\right)}_{{m}{+}{n}}{}{m}{!}{}{n}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__1}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__2}{;}{c}{+}{k}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__2}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__1}{;}{c}{+}{k}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__2}\right|{<}{1}\right]$ (9)

As indicated in the formulas above, for AppellF1 (also for AppellF3) the domain of convergence of the single sum with hypergeometric coefficients is larger than the domain of convergence of the double series, because the hypergeometric coefficient in the single sum - say the one in ${z}_{2}$ - analytically extends the series with regards to the other variable - say ${z}_{1}$ - entering the hypergeometric coefficient. Hence, for AppellF1 (also for AppellF3), the case where one of the two variables, ${z}_{1}$ or ${z}_{2}$, is equal to 1, is convergent only when the corresponding hypergeometric coefficient in the single sum form is convergent. For instance, the convergent case at ${z}_{1}=1$ requires that .

AppellF1 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,

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

Among other situations, this identity is useful when both ${z}_{1}$ and ${z}_{2}$ have absolute values larger than 1 but one of the arguments in the same position of AppellF1 on the right-hand side has absolute value smaller than 1.

A contiguity transformation for AppellF1

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

The contiguity transformations available in this way are

 > $\mathrm{indices}\left(\mathrm{AppellF1}:-\mathrm{Transformations}\left["Contiguity"\right]\right)$
 $\left[{1}\right]{,}\left[{2}\right]{,}\left[{3}\right]{,}\left[{4}\right]{,}\left[{5}\right]{,}\left[{6}\right]{,}\left[{7}\right]{,}\left[{9}\right]{,}\left[{8}\right]{,}\left[{10}\right]$ (12)

By using differential algebra techniques, the PDE system satisfied by AppellF1 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 AppellF1 this linear ODE is of third order and can be computed as follows

 >
 ${\mathrm{F1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ (13)
 >
 $\frac{{{\partial }}^{{3}}}{{\partial }{\mathrm{z__2}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\left(\left({a}{+}{2}{}\mathrm{b__2}{+}{4}\right){}{\mathrm{z__2}}^{{2}}{+}\left(\left({-}{a}{+}\mathrm{b__1}{-}\mathrm{b__2}{-}{3}\right){}\mathrm{z__1}{-}{c}{-}\mathrm{b__2}{-}{2}\right){}\mathrm{z__2}{+}\mathrm{z__1}{}\left({c}{-}\mathrm{b__1}{+}{1}\right)\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__2}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right){+}\left(\left(\left({2}{}{a}{+}\mathrm{b__2}{+}{2}\right){}\mathrm{z__2}{+}\left({-}{a}{+}\mathrm{b__1}{-}{1}\right){}\mathrm{z__1}{-}{c}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right){+}{\mathrm{F1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}{}\mathrm{b__2}\right){}\left(\mathrm{b__2}{+}{1}\right)}{\left({-}{1}{+}\mathrm{z__2}\right){}\left({-}\mathrm{z__2}{+}\mathrm{z__1}\right){}\mathrm{z__2}}$ (14)

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

 > $\mathrm{DEtools}\left[\mathrm{singularities}\right]\left(\mathrm{subs}\left(\mathrm{F1}\left(\mathrm{z__1},\mathrm{z__2}\right)=\mathrm{F1}\left(\mathrm{z__2}\right),\right)\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}{,}\mathrm{z__1}{,}{\mathrm{\infty }}\right\}{,}{\mathrm{irregular}}{=}{\varnothing }$ (15)

You can also see a general presentation of AppellF1, organized into sections and including plots, using the FunctionAdvisor

 > $\mathrm{FunctionAdvisor}\left(\mathrm{AppellF1}\right)$

AppellF1

describe

 ${\mathrm{AppellF1}}{=}{\mathrm{Appell 2-variable hypergeometric function F1}}$

definition

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

classify function

 ${\mathrm{Appell}}$

symmetries

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$

plot

singularities

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left({c}{::}{\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}{<}{c}\right){\vee }\left({c}{::}{\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}{<}{c}\right){\vee }\left({c}{::}{\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 }\mathrm{b__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{\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({c}{::}{\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({c}{::}{\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__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right)$

branch points

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

branch cuts

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

special values

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

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

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

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

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

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

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

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

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

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

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

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{1}}{F}_{{0}}{}\left(\mathrm{b__1}{;}{;}\mathrm{z__1}\right){}{}_{{1}}{F}_{{0}}{}\left(\mathrm{b__2}{;}{;}\mathrm{z__2}\right)$ ${c}{=}{a}{\wedge }{a}{\ne }{0}$

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

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{arctanh}}{}\left(\sqrt{\mathrm{z__1}}\right){}\sqrt{\mathrm{z__1}}{-}{\mathrm{arctanh}}{}\left(\sqrt{\mathrm{z__2}}\right){}\sqrt{\mathrm{z__2}}}{{-}\mathrm{z__2}{+}\mathrm{z__1}}$ ${a}{=}\frac{{1}}{{2}}{\wedge }\mathrm{b__1}{=}{1}{\wedge }\mathrm{b__2}{=}{1}{\wedge }{c}{=}\frac{{3}}{{2}}$

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{3}{}\left(\frac{{\mathrm{arctanh}}{}\left(\frac{\sqrt{\mathrm{z__1}}{}\sqrt{\frac{\mathrm{z__2}{-}\mathrm{z__1}}{\mathrm{z__1}}}}{\sqrt{{1}{-}\mathrm{z__1}}}\right)}{\sqrt{\frac{\mathrm{z__2}{-}\mathrm{z__1}}{\mathrm{z__1}}}}{-}{\mathrm{arcsin}}{}\left(\sqrt{\mathrm{z__1}}\right)\right)}{\sqrt{\mathrm{z__1}}{}\mathrm{z__2}}$ ${a}{=}\frac{{3}}{{2}}{\wedge }\mathrm{b__1}{=}\frac{{1}}{{2}}{\wedge }\mathrm{b__2}{=}{1}{\wedge }{c}{=}\frac{{5}}{{2}}$

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{2}{}{a}{}\left(\frac{{\mathrm{arctanh}}{}\left(\frac{\sqrt{\mathrm{z__1}}{}\sqrt{{-}{1}{+}\frac{\mathrm{z__2}}{\mathrm{z__1}}}}{\sqrt{{1}{-}\mathrm{z__1}}}\right)}{{\mathrm{z__2}}^{{-}\frac{{1}}{{2}}{+}{a}}{}\sqrt{{-}{1}{+}\frac{\mathrm{z__2}}{\mathrm{z__1}}}}{-}\frac{{\sum }_{{\mathrm{_k2}}{=}{0}}^{{-}\frac{{3}}{{2}}{+}{a}}{}{\left({-1}\right)}^{{\mathrm{_k2}}}{}\left(\genfrac{}{}{0}{}{{-}\frac{{1}}{{2}}{+}{a}}{{\mathrm{_k2}}{+}{1}}\right){}{\left(\frac{{-}{2}{}\mathrm{z__1}{+}\mathrm{z__2}}{\mathrm{z__2}}\right)}^{{\mathrm{_k2}}{+}{1}}{}\left({\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{_k2}}{+}{1}}{}\frac{\left(\genfrac{}{}{0}{}{{\mathrm{_k2}}{+}{1}}{{\mathrm{_k1}}}\right){}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}\left(\left({\sum }_{{\mathrm{_k4}}{=}{0}}^{⌊\frac{{\mathrm{_k1}}}{{2}}{+}\frac{{1}}{{2}}⌋{-}{1}}{}\frac{\left(\genfrac{}{}{0}{}{{\mathrm{_k1}}{-}{1}}{{2}{}{\mathrm{_k4}}}\right){}{\left(\frac{{1}}{{2}}\right)}_{{\mathrm{_k4}}}{}{\left(\frac{{2}{}\mathrm{z__1}{-}\mathrm{z__2}}{\mathrm{z__1}}\right)}^{{\mathrm{_k1}}{-}{1}{-}{2}{}{\mathrm{_k4}}}{}{\mathrm{z__2}}^{{2}{}{\mathrm{_k4}}}{}\left({2}{}{\mathrm{arcsin}}{}\left(\sqrt{\mathrm{z__1}}\right){+}\sqrt{{1}{-}\mathrm{z__1}}{}\sqrt{\mathrm{z__1}}{}\left({\sum }_{{\mathrm{_k3}}{=}{1}}^{{\mathrm{_k4}}}{}\frac{\left({\mathrm{_k3}}{-}{1}\right){!}{}{\left({1}{-}{2}{}\mathrm{z__1}\right)}^{{2}{}{\mathrm{_k3}}{-}{1}}}{{\left(\frac{{1}}{{2}}\right)}_{{\mathrm{_k3}}}}\right)\right)}{{\mathrm{_k4}}{!}{}{\mathrm{z__1}}^{{2}{}{\mathrm{_k4}}}}\right){+}\left({\sum }_{{\mathrm{_k4}}{=}{0}}^{{-}{1}{+}⌊\frac{{\mathrm{_k1}}}{{2}}⌋}{}{{2}}^{{2}{}{\mathrm{_k4}}{+}{1}}{}\left(\genfrac{}{}{0}{}{{\mathrm{_k1}}{-}{1}}{{2}{}{\mathrm{_k4}}{+}{1}}\right){}{\mathrm{_k4}}{!}{}{\left({1}{-}\mathrm{z__1}\right)}^{{\mathrm{_k4}}{+}\frac{{1}}{{2}}}{}{\mathrm{z__1}}^{{-}\frac{{1}}{{2}}{-}{\mathrm{_k4}}}{}{\left(\frac{{2}{}\mathrm{z__1}{-}\mathrm{z__2}}{\mathrm{z__1}}\right)}^{{\mathrm{_k1}}{-}{2}{-}{2}{}{\mathrm{_k4}}}{}{\mathrm{z__2}}^{{2}{}{\mathrm{_k4}}{+}{1}}{}\left({\sum }_{{\mathrm{_k3}}{=}{0}}^{{\mathrm{_k4}}}{}\frac{{\left({1}{-}{2}{}\mathrm{z__1}\right)}^{{2}{}{\mathrm{_k3}}}}{{{2}}^{{2}{}{\mathrm{_k3}}}{}{\left({1}{-}\mathrm{z__1}\right)}^{{\mathrm{_k3}}}{}{\mathrm{z__1}}^{{\mathrm{_k3}}}{}{\mathrm{_k3}}{!}{}{\left(\frac{{3}}{{2}}\right)}_{{\mathrm{_k4}}{-}{\mathrm{_k3}}}}\right)\right)\right)}{{\left({-}{2}{}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{\mathrm{_k1}}}}\right)}{{{2}}^{{-}\frac{{1}}{{2}}{+}{a}}{}{\mathrm{z__1}}^{{-}\frac{{1}}{{2}}{+}{a}}}\right)}{\sqrt{\mathrm{z__1}}}$ $\left({-}\frac{{1}}{{2}}{+}{a}\right){::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonnegint}}{'}\right]\right){\wedge }\mathrm{b__1}{=}\frac{{1}}{{2}}{\wedge }\mathrm{b__2}{=}{1}{\wedge }{c}{-}{1}{-}{a}{=}{0}$

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{3}{}\left(\sqrt{\mathrm{z__1}}{}\left(\mathrm{z__1}{-}{2}{}{\mathrm{arctanh}}{}\left(\sqrt{\mathrm{z__2}}\right){}\left({-}{1}{+}\mathrm{z__1}\right){}\sqrt{\mathrm{z__2}}{-}\mathrm{z__2}\right){+}{\mathrm{arctanh}}{}\left(\sqrt{\mathrm{z__1}}\right){}\left({-}{1}{+}\mathrm{z__1}\right){}\left(\mathrm{z__2}{+}\mathrm{z__1}\right)\right)}{{2}{}\left({1}{-}\mathrm{z__1}\right){}\sqrt{\mathrm{z__1}}{}{\left({-}\mathrm{z__2}{+}\mathrm{z__1}\right)}^{{2}}}$ ${a}{=}\frac{{3}}{{2}}{\wedge }\mathrm{b__1}{=}{2}{\wedge }\mathrm{b__2}{=}{1}{\wedge }{c}{=}\frac{{5}}{{2}}$

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{5}{}\left({\mathrm{arctanh}}{}\left(\sqrt{\mathrm{z__1}}\right){}{\left({-}{1}{+}\mathrm{z__1}\right)}^{{2}}{}\left({3}{}{\mathrm{z__1}}^{{2}}{+}{6}{}\mathrm{z__1}{}\mathrm{z__2}{-}{\mathrm{z__2}}^{{2}}\right){+}\sqrt{\mathrm{z__1}}{}\left({5}{}{\mathrm{z__1}}^{{3}}{-}{8}{}{\mathrm{arctanh}}{}\left(\sqrt{\mathrm{z__2}}\right){}{\left({-}{1}{+}\mathrm{z__1}\right)}^{{2}}{}\mathrm{z__1}{}\sqrt{\mathrm{z__2}}{+}{\mathrm{z__2}}^{{2}}{+}\mathrm{z__1}{}\mathrm{z__2}{}\left({2}{+}\mathrm{z__2}\right){-}{3}{}{\mathrm{z__1}}^{{2}}{}\left({1}{+}{2}{}\mathrm{z__2}\right)\right)\right)}{{8}{}{\left({-}{1}{+}\mathrm{z__1}\right)}^{{2}}{}{\mathrm{z__1}}^{{3}}{{2}}}{}{\left({-}\mathrm{z__2}{+}\mathrm{z__1}\right)}^{{3}}}$ ${a}{=}\frac{{5}}{{2}}{\wedge }\mathrm{b__1}{=}{3}{\wedge }\mathrm{b__2}{=}{1}{\wedge }{c}{=}\frac{{7}}{{2}}$

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\left({2}{}{c}{-}{2}\right){}\left({\left({-1}\right)}^{{-}\frac{{5}}{{2}}{+}{c}}{}{\left({1}{-}\mathrm{z__2}\right)}^{{-}\frac{{3}}{{2}}{+}{c}}{}{\mathrm{z__2}}^{\frac{{1}}{{2}}{-}{c}}{}\sqrt{\frac{\mathrm{z__2}{-}\mathrm{z__1}}{{-}{1}{+}\mathrm{z__2}}}{}{\mathrm{arctanh}}{}\left(\sqrt{\frac{\mathrm{z__2}{-}\mathrm{z__1}}{{-}{1}{+}\mathrm{z__2}}}\right){+}\frac{{\left({-1}\right)}^{{-}\frac{{3}}{{2}}{+}{c}}{}{\left({1}{-}\mathrm{z__1}\right)}^{{c}{-}\frac{{1}}{{2}}}{}\left({\sum }_{{\mathrm{_k1}}{=}{0}}^{{-}\frac{{5}}{{2}}{+}{c}}{}\frac{\left({\mathrm{_k1}}{+}{1}\right){}{{2}}^{{\mathrm{_k1}}{-}{2}{}{c}{+}{3}}{}{\mathrm{z__1}}^{\frac{{\mathrm{_k1}}}{{2}}{+}{2}{-}{c}}{}\left({2}{}{c}{-}{5}{-}{\mathrm{_k1}}\right){!}{}\left(\left({\left(\sqrt{\mathrm{z__1}}{-}\sqrt{\frac{\mathrm{z__2}{-}\mathrm{z__1}}{{-}{1}{+}\mathrm{z__2}}}\right)}^{{-}{2}{-}{\mathrm{_k1}}}{+}{\left(\sqrt{\mathrm{z__1}}{+}\sqrt{\frac{\mathrm{z__2}{-}\mathrm{z__1}}{{-}{1}{+}\mathrm{z__2}}}\right)}^{{-}{2}{-}{\mathrm{_k1}}}\right){}{\mathrm{arctanh}}{}\left(\sqrt{\mathrm{z__1}}\right){-}\left({\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{_k1}}}{}\frac{{\left({-1}\right)}^{{\mathrm{_k2}}}{}\left({\left(\sqrt{\mathrm{z__1}}{-}\sqrt{\frac{\mathrm{z__2}{-}\mathrm{z__1}}{{-}{1}{+}\mathrm{z__2}}}\right)}^{{\mathrm{_k2}}{-}{\mathrm{_k1}}{-}{1}}{+}{\left(\sqrt{\mathrm{z__1}}{+}\sqrt{\frac{\mathrm{z__2}{-}\mathrm{z__1}}{{-}{1}{+}\mathrm{z__2}}}\right)}^{{\mathrm{_k2}}{-}{\mathrm{_k1}}{-}{1}}\right){}\left({\sum }_{{\mathrm{_k3}}{=}{0}}^{{\mathrm{_k2}}}{}\frac{{{2}}^{{\mathrm{_k2}}{-}{2}{}{\mathrm{_k3}}}{}{\left({1}{-}\mathrm{z__1}\right)}^{{-}{1}{-}{\mathrm{_k2}}{+}{\mathrm{_k3}}}{}{\mathrm{z__1}}^{\frac{{\mathrm{_k2}}}{{2}}{-}{\mathrm{_k3}}}{}\left({\mathrm{_k2}}{-}{\mathrm{_k3}}\right){!}}{\left({\mathrm{_k2}}{-}{2}{}{\mathrm{_k3}}\right){!}{}{\mathrm{_k3}}{!}}\right)}{{\mathrm{_k2}}{+}{1}}\right)\right)}{\left({c}{-}\frac{{5}}{{2}}{-}{\mathrm{_k1}}\right){!}}\right)}{\left({-}\frac{{3}}{{2}}{+}{c}\right){!}{}\left({1}{-}\mathrm{z__2}\right)}\right)$ ${a}{=}{1}{\wedge }\mathrm{b__1}{=}{-}\frac{{1}}{{2}}{\wedge }\mathrm{b__2}{=}{1}{\wedge }\left({-}\frac{{3}}{{2}}{+}{c}\right){::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonnegint}}{'}\right]\right)$

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\left\{\begin{array}{cc}{1}& \left(\mathrm{b__1}{=}{0}{\wedge }\mathrm{b__2}{=}{0}\right){\vee }{a}{=}{0}\\ \frac{{\sum }_{{\mathrm{_k2}}{=}{0}}^{{a}{-}{c}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{a}{-}{c}{-}{\mathrm{_k2}}}{}\frac{{\left({-}{a}{+}{c}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k2}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k1}}}{}{\left(\frac{\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}}\right)}^{{\mathrm{_k2}}}{}{\left(\frac{\mathrm{z__2}}{{-}{1}{+}\mathrm{z__2}}\right)}^{{\mathrm{_k1}}}}{{\left({c}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{_k2}}{!}{}{\mathrm{_k1}}{!}}}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__1}}{}{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{b__2}}}& {\mathrm{otherwise}}\end{array}\right\$ $\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__2}{\ne }{1}{\wedge }\left({a}{-}{c}\right){::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonnegint}}{'}\right]\right)$

identities

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

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

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

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

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left({a}\right)}_{{n}}{}{{F}}_{{1}}{}\left({n}{+}{a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left({a}{-}{c}{+}{1}\right)}_{{n}}}{-}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{{\left({-1}\right)}^{{k}}{}\left(\genfrac{}{}{0}{}{{n}}{{k}}\right){}{\left({1}{-}{c}\right)}_{{k}}{}{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{-}{k}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left({a}{-}{c}{+}{1}\right)}_{{k}}}\right)$ ${c}{\ne }{1}{\wedge }\left(\left({a}{-}{c}{+}{1}\right){::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\vee }{n}{\le }\left|{a}{-}{c}{+}{1}\right|\right)$

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

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

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

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

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

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

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

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

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

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

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{Physics}}{:-}{\mathrm{Library}}{:-}{\mathrm{Add}}{}\left(\frac{{\left({a}\right)}_{{\mathrm{k1}}{+}{\mathrm{k2}}}{}{{F}}_{{3}}{}\left(\mathrm{b__1}{,}\mathrm{b__2}{,}{-}{\mathrm{k1}}{,}{-}{\mathrm{k2}}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\mathrm{k1}}{!}{}{\mathrm{k2}}{!}}{,}{\mathrm{k1}}{+}{\mathrm{k2}}{\le }{-}{a}\right)}{{\left({-1}\right)}^{{a}}}$ ${a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)$

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

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

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left(\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}^{\mathrm{b__2}}{}{\left({1}{+}\sqrt{\frac{{4}{}\mathrm{z__2}{}\left(\mathrm{z__2}{+}\mathrm{z__1}\right){}\sqrt{\frac{\mathrm{z__1}}{\mathrm{z__2}}}{+}{\mathrm{z__1}}^{{2}}{+}{6}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\mathrm{z__2}}^{{2}}}{{\left(\mathrm{z__1}{-}\mathrm{z__2}\right)}^{{2}}}}\right)}^{{2}{}\mathrm{b__1}{+}{2}{}\mathrm{b__2}}{}{{F}}_{{4}}{}\left({a}{,}\mathrm{b__1}{+}\mathrm{b__2}{,}\mathrm{b__1}{+}\mathrm{b__2}{-}{a}{+}{1}{,}{c}{,}\frac{{4}{}\mathrm{z__2}{}\left(\mathrm{z__2}{+}\mathrm{z__1}\right){}\sqrt{\frac{\mathrm{z__1}}{\mathrm{z__2}}}{+}{\mathrm{z__1}}^{{2}}{+}{6}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\mathrm{z__2}}^{{2}}}{{\left(\mathrm{z__1}{-}\mathrm{z__2}\right)}^{{2}}}{,}\frac{{2}{}\mathrm{z__1}{}\left({\left(\mathrm{z__1}{-}\mathrm{z__2}\right)}^{{2}}{}\sqrt{\frac{{4}{}\mathrm{z__2}{}\left(\mathrm{z__2}{+}\mathrm{z__1}\right){}\sqrt{\frac{\mathrm{z__1}}{\mathrm{z__2}}}{+}{\mathrm{z__1}}^{{2}}{+}{6}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\mathrm{z__2}}^{{2}}}{{\left(\mathrm{z__1}{-}\mathrm{z__2}\right)}^{{2}}}}{+}\left(\mathrm{z__2}{+}\mathrm{z__1}\right){}\left({2}{}\sqrt{\frac{\mathrm{z__1}}{\mathrm{z__2}}}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)\right)}{{\left(\mathrm{z__1}{-}\mathrm{z__2}\right)}^{{2}}}\right)$ ${a}{-}\mathrm{b__1}{-}\frac{{1}}{{2}}{=}{0}{\wedge }{-}\mathrm{b__2}{+}\mathrm{b__1}{=}{0}{\wedge }{{\mathrm{\Re }}{}\left(\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}^{{2}}{+}{{\mathrm{\Im }}{}\left(\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}^{{2}}{\le }{1}$

 ${{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left(\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}^{\mathrm{b__2}}{}{\left({1}{+}\sqrt{\frac{\left({-}{4}{}\mathrm{z__1}{+}{8}\right){}\sqrt{{1}{-}\mathrm{z__1}}{+}{\mathrm{z__1}}^{{2}}{-}{8}{}\mathrm{z__1}{+}{8}}{{\mathrm{z__1}}^{{2}}}}\right)}^{{2}{}\mathrm{b__1}{+}{2}{}\mathrm{b__2}}{}{{F}}_{{4}}{}\left(\mathrm{b__2}{,}\mathrm{b__1}{+}\mathrm{b__2}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__1}{+}\mathrm{b__2}{,}\frac{\left({-}{4}{}\mathrm{z__1}{+}{8}\right){}\sqrt{{1}{-}\mathrm{z__1}}{+}{\mathrm{z__1}}^{{2}}{-}{8}{}\mathrm{z__1}{+}{8}}{{\mathrm{z__1}}^{{2}}}{,}{-}\frac{{2}{}\left(\sqrt{\frac{\left({-}{4}{}\mathrm{z__1}{+}{8}\right){}\sqrt{{1}{-}\mathrm{z__1}}{+}{\mathrm{z__1}}^{{2}}{-}{8}{}\mathrm{z__1}{+}{8}}{{\mathrm{z__1}}^{{2}}}}{}{\mathrm{z__1}}^{{2}}{+}\left(\mathrm{z__1}{-}{2}\right){}\left(\mathrm{z__1}{-}{2}{}\sqrt{{1}{-}\mathrm{z__1}}{-}{2}\right)\right){}\left(\mathrm{z__1}{-}\mathrm{z__2}\right)}{{\mathrm{z__1}}^{{2}}{}\mathrm{z__2}}\right)$ ${a}{-}\mathrm{b__1}{-}\frac{{1}}{{2}}{=}{0}{\wedge }{c}{-}{2}{}{a}{=}{0}{\wedge }{\left({\mathrm{\Re }}{}\left(\mathrm{z__1}\right){-}{1}\right)}^{{2}}{+}{{\mathrm{\Im }}{}\left(\mathrm{z__1}\right)}^{{2}}{\le }{1}$

sum form

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

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

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

series

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

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

integral form

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

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

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

differentiation rule

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

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

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

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

DE

${f}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$

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