AppellF3 - Maple Help

AppellF3

The AppellF3 function

 Calling Sequence AppellF3(${a}_{1},{a}_{2},{b}_{1},{b}_{2},c,{z}_{1},{z}_{2}$)

Parameters

 ${a}_{1}$ - algebraic expression ${a}_{2}$ - 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, AppellF3, 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, AppellF3 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 AppellF3 series and the corresponding domain of convergence can be seen through the FunctionAdvisor
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{a__2}\right)}_{{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left({c}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]$ (1)
 A distinction is made between the AppellF3 doubly hypergeometric series, with the restricted domain of convergence shown above, and the AppellF3 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 AppellF3 variables subscripted with the numbers 1 and 2, the function remains the same; hence
 > $\mathrm{FunctionAdvisor}\left(\mathrm{symmetries},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__2}{,}\mathrm{a__1}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}{{F}}_{{3}}{}\left(\mathrm{b__1}{,}\mathrm{a__2}{,}\mathrm{a__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}\mathrm{a__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right]$ (2)
 Note the existence of other symmetries, also visible in the double sum definition.
 From the series' definition, AppellF3 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 ${a}_{1}$ and/or ${a}_{2}$ and/or ${b}_{1}$ and/or ${b}_{2}$ parameters entering the pochhammer functions in the numerator of the series are non-positive integers, the series will truncate and AppellF3 will be polynomial in one of the two of ${z}_{1},{z}_{2}$. As is the case of the hypergeometric function, when the pochhammers in both the numerator and the denominator have non-positive integer arguments, AppellF3 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
 > $\mathrm{FunctionAdvisor}\left(\mathrm{singularities},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{a__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{+}\mathrm{a__2}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{a__2}{::}\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){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{a__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{a__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{a__2}{<}{c}\right)\right]$ (3)
 The AppellF3 series is analytically extended to the AppellF3 function defined over the whole complex plane using identities and mainly by integral representations in terms of Eulerian integrals:
 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral_form},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\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}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{-}\mathrm{b__1}{;}\mathrm{z__2}{}{u}\right)}{{{u}}^{{-}{c}{+}\mathrm{b__1}{+}{1}}{}{\left({1}{+}\left({u}{-}{1}\right){}\mathrm{z__1}\right)}^{\mathrm{a__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)}{,}{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({c}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}{c}{+}\mathrm{b__1}\right)\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\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}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{-}\mathrm{b__2}{;}\mathrm{z__1}{}{u}\right)}{{{u}}^{{-}{c}{+}\mathrm{b__2}{+}{1}}{}{\left({1}{+}\left({u}{-}{1}\right){}\mathrm{z__2}\right)}^{\mathrm{a__2}}}\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)}{,}{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({c}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}{c}{+}\mathrm{b__2}\right)\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}{{u}}^{{\mathrm{\rho }}{-}{1}}{}{\left({1}{-}{u}\right)}^{{c}{-}{\mathrm{\rho }}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{\mathrm{\rho }}{;}\mathrm{z__1}{}{u}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{-}{\mathrm{\rho }}{;}{-}\left({u}{-}{1}\right){}\mathrm{z__2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left({\mathrm{\rho }}\right){}{\mathrm{\Gamma }}{}\left({c}{-}{\mathrm{\rho }}\right)}{,}{0}{<}{\mathrm{\Re }}{}\left({c}\right)\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\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({-}\mathrm{z__1}{}{u}{+}{1}\right)}^{\mathrm{a__1}}{}{\left({-}{v}{}\mathrm{z__2}{+}{1}\right)}^{\mathrm{a__2}}}\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({c}{-}\mathrm{b__1}{-}\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 AppellF3.
 AppellF3 also satisfies a linear system of partial differential equations of second order
 > $\mathrm{FunctionAdvisor}\left(\mathrm{DE},\mathrm{AppellF3}\right)$
 $\left[{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\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(\mathrm{a__2}{,}\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(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{+}\frac{\left(\left({-}\mathrm{a__1}{-}\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(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{-}\frac{\mathrm{a__1}{}\mathrm{b__1}{}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\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(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{z__2}{}\left(\mathrm{z__2}{-}{1}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__2}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}}{+}\frac{\left(\left(\mathrm{a__2}{+}\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(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}}{+}\frac{\mathrm{a__2}{}\mathrm{b__2}{}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\mathrm{z__1}}\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 AppellF3. For example, the twelve polynomial cases of AppellF3 are

 > $\mathrm{AppellF3}:-\mathrm{SpecialValues}:-\mathrm{Polynomial}\left(\right)$
 ${12}{,}\left({\mathrm{a1}}{,}{\mathrm{a2}}{,}{\mathrm{b1}}{,}{\mathrm{b2}}{,}{c}{,}{\mathrm{z1}}{,}{\mathrm{z2}}\right){↦}\left[\left[{\mathrm{a1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{a2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{a1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{a2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{a1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{a2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{\le }{\mathrm{a1}}{+}{\mathrm{a2}}\right]{,}\left[{\mathrm{a1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{\le }{\mathrm{a1}}{+}{\mathrm{b2}}\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{a2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{\le }{\mathrm{b1}}{+}{\mathrm{a2}}\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{\le }{\mathrm{b1}}{+}{\mathrm{b2}}\right]{,}\left[{\mathrm{a1}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{a2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]\right]$ (6)

Likewise, the conditions for the singular cases of AppellF3 can be seen either using the FunctionAdvisor or entering AppellF3:-Singularities(), so with no arguments.

For particular values of its parameters, AppellF3 is related to the hypergeometric function. These hypergeometric cases are returned automatically. For example, for ${z}_{1}=1$,

 > $\left(\mathrm{%AppellF3}=\mathrm{AppellF3}\right)\left(\mathrm{a__1},\mathrm{a__2},\mathrm{b__1},\mathrm{b__2},c,1,\mathrm{z__2}\right)$
 ${\mathrm{%AppellF3}}{}\left(\mathit{a__1}{,}\mathit{a__2}{,}\mathit{b__1}{,}\mathit{b__2}{,}{c}{,}{1}{,}\mathit{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{;}{1}\right){}{}_{{3}}{F}_{{2}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{,}{c}{-}\mathrm{a__1}{-}\mathrm{b__1}{;}{c}{-}\mathrm{b__1}{,}{c}{-}\mathrm{a__1}{;}\mathrm{z__2}\right)$ (7)

This formula analytically extends to the whole complex plane the AppellF3 series when any of ${z}_{1}=1$ or ${z}_{2}=1$ (the latter using the symmetry of AppellF3 - see the beginning of the Description section).

To see all the hypergeometric cases, enter

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{AppellF3},\mathrm{hypergeom}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{;}\mathrm{z__2}\right){,}\mathrm{z__1}{=}{0}{\vee }\mathrm{a__1}{=}{0}{\vee }\mathrm{b__1}{=}{0}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{;}\mathrm{z__1}\right){,}\mathrm{z__2}{=}{0}{\vee }\mathrm{a__2}{=}{0}{\vee }\mathrm{b__2}{=}{0}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{;}{1}\right){}{}_{{3}}{F}_{{2}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{,}{c}{-}\mathrm{a__1}{-}\mathrm{b__1}{;}{c}{-}\mathrm{b__1}{,}{c}{-}\mathrm{a__1}{;}\mathrm{z__2}\right){,}\mathrm{z__1}{=}{1}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{;}{1}\right){}{}_{{3}}{F}_{{2}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{,}{c}{-}\mathrm{a__2}{-}\mathrm{b__2}{;}{c}{-}\mathrm{b__2}{,}{c}{-}\mathrm{a__2}{;}\mathrm{z__1}\right){,}\mathrm{z__2}{=}{1}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{-}\mathrm{z__1}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{a__1}{;}\mathrm{a__1}{+}\mathrm{a__2}{;}\mathrm{z__1}\right){-}\mathrm{z__2}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{a__2}{;}\mathrm{a__1}{+}\mathrm{a__2}{;}\mathrm{z__2}\right)}{\left(\mathrm{z__2}{-}{1}\right){}\mathrm{z__1}{-}\mathrm{z__2}}{,}\mathrm{b__1}{=}{1}{\wedge }\mathrm{b__2}{=}{1}{\wedge }{c}{=}\mathrm{a__1}{+}\mathrm{a__2}{\wedge }{-}\mathrm{z__1}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}{\ne }{0}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{-}\mathrm{z__1}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{b__1}{;}\mathrm{b__1}{+}\mathrm{a__2}{;}\mathrm{z__1}\right){-}\mathrm{z__2}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{a__2}{;}\mathrm{b__1}{+}\mathrm{a__2}{;}\mathrm{z__2}\right)}{\left(\mathrm{z__2}{-}{1}\right){}\mathrm{z__1}{-}\mathrm{z__2}}{,}\mathrm{a__1}{=}{1}{\wedge }\mathrm{b__2}{=}{1}{\wedge }{c}{=}\mathrm{b__1}{+}\mathrm{a__2}{\wedge }{-}\mathrm{z__1}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}{\ne }{0}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{-}\mathrm{z__1}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{a__1}{;}\mathrm{a__1}{+}\mathrm{b__2}{;}\mathrm{z__1}\right){-}\mathrm{z__2}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{b__2}{;}\mathrm{a__1}{+}\mathrm{b__2}{;}\mathrm{z__2}\right)}{\left(\mathrm{z__2}{-}{1}\right){}\mathrm{z__1}{-}\mathrm{z__2}}{,}\mathrm{b__1}{=}{1}{\wedge }\mathrm{a__2}{=}{1}{\wedge }{c}{=}\mathrm{a__1}{+}\mathrm{b__2}{\wedge }{-}\mathrm{z__1}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}{\ne }{0}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__2}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{+}\mathrm{b__2}{,}\mathrm{b__1}{+}\mathrm{b__2}{;}\mathrm{a__1}{+}\mathrm{a__2}{+}\mathrm{b__1}{+}\mathrm{b__2}{;}\mathrm{z__1}\right){,}\left({c}{=}\mathrm{a__1}{+}\mathrm{a__2}{+}\mathrm{b__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__2}{=}\frac{\mathrm{z__1}}{\mathrm{z__1}{-}{1}}\right){\vee }\left({c}{=}\mathrm{a__1}{+}\mathrm{a__2}{+}\mathrm{b__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__2}{\ne }{1}{\wedge }\mathrm{z__1}{=}\frac{\mathrm{z__2}}{\mathrm{z__2}{-}{1}}\right)\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left({1}{-}\mathrm{z__1}\right)}^{{c}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left(\frac{{c}}{{2}}{+}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{,}\frac{{c}}{{2}}{-}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{+}\frac{{1}}{{2}}{;}{c}{;}{-}{4}{}\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)\right)}{{\left({1}{-}{2}{}\mathrm{z__1}\right)}^{{-}{1}{+}\mathrm{a__2}}}{,}\left(\mathrm{z__2}{=}\frac{\mathrm{z__1}}{{2}{}\mathrm{z__1}{-}{1}}{\wedge }{2}{}\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{b__1}{=}{1}{-}\mathrm{a__1}{\wedge }\mathrm{b__2}{=}{1}{-}\mathrm{a__2}\right){\vee }\left(\mathrm{z__1}{=}\frac{\mathrm{z__2}}{{2}{}\mathrm{z__2}{-}{1}}{\wedge }{2}{}\mathrm{z__2}{\ne }{1}{\wedge }\mathrm{z__2}{\ne }{1}{\wedge }\mathrm{b__1}{=}{1}{-}\mathrm{a__1}{\wedge }\mathrm{b__2}{=}{1}{-}\mathrm{a__2}\right){\vee }\left(\mathrm{z__2}{=}\frac{\mathrm{z__1}}{{2}{}\mathrm{z__1}{-}{1}}{\wedge }{2}{}\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{a__1}{=}{1}{-}\mathrm{b__1}{\wedge }\mathrm{b__2}{=}{1}{-}\mathrm{a__2}\right){\vee }\left(\mathrm{z__2}{=}\frac{\mathrm{z__1}}{{2}{}\mathrm{z__1}{-}{1}}{\wedge }{2}{}\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{b__1}{=}{1}{-}\mathrm{a__1}{\wedge }\mathrm{a__2}{=}{1}{-}\mathrm{b__2}\right)\right]$ (8)

Other special values of AppellF3 can be seen using FunctionAdvisor(special_values, AppellF3).

By requesting the sum form of AppellF3, 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:

 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{m}}{}{\left(\mathrm{a__2}\right)}_{{n}}{}{\left(\mathrm{b__1}\right)}_{{m}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left({c}\right)}_{{m}{+}{n}}{}{m}{!}{}{n}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{k}}{}{\left(\mathrm{b__1}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{+}{k}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__2}\right)}_{{k}}{}{\left(\mathrm{b__2}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{+}{k}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__2}\right|{<}{1}\right]$ (9)

As indicated in the formulas above, for AppellF3 (also for AppellF1) 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 AppellF3 (also for AppellF1), 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 $0<\mathrm{Re}\left(c-{a}_{1}-{b}_{1}\right)$.

AppellF3 is the only one of the four Appell functions that does not admit identities analogous to the Euler identities for the hypergeometric function.

A contiguity transformation for AppellF3

 >
 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{+}{n}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{\left(\genfrac{}{}{0}{}{{n}}{{k}}\right){}{\left(\mathrm{a__2}\right)}_{{k}}{}{\mathrm{z__2}}^{{k}}{}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{b__2}{+}{k}{,}\mathrm{b__1}{,}{k}{+}\mathrm{a__2}{,}{k}{+}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left({c}\right)}_{{k}}}\right){,}\mathrm{z__2}{\ne }{1}{\wedge }\left({c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\vee }\left(\mathrm{a__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{c}{<}\mathrm{a__2}\right){\vee }{n}{\le }\left|{c}\right|\right)\right]$ (10)

The contiguity transformations available in this way are

 > $\mathrm{indices}\left(\mathrm{AppellF3}:-\mathrm{Transformations}\left["Contiguity"\right]\right)$
 $\left[{1}\right]{,}\left[{2}\right]{,}\left[{3}\right]{,}\left[{4}\right]{,}\left[{5}\right]{,}\left[{6}\right]$ (11)

By using differential algebra techniques, the PDE system satisfied by AppellF3 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 AppellF3 this linear ODE is of fourth order and can be computed as follows

 >
 ${\mathrm{F3}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ (12)
 >
 $\frac{{{\partial }}^{{4}}}{{\partial }{\mathrm{z__2}}^{{4}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F3}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{2}{}\mathrm{z__2}{}\left({-}\left(\mathrm{z__1}{-}{1}\right){}\left(\mathrm{a__2}{+}\mathrm{b__2}{+}{4}\right){}{\mathrm{z__2}}^{{2}}{+}\left(\left({c}{-}\frac{\mathrm{a__1}}{{2}}{+}\mathrm{a__2}{-}\frac{\mathrm{b__1}}{{2}}{+}\mathrm{b__2}{+}\frac{{11}}{{2}}\right){}\mathrm{z__1}{-}\frac{{c}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{-}\frac{\mathrm{b__2}}{{2}}{-}\frac{{5}}{{2}}\right){}\mathrm{z__2}{-}\left({c}{-}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{b__1}}{{2}}{+}\frac{{3}}{{2}}\right){}\mathrm{z__1}\right){}\left(\frac{{{\partial }}^{{3}}}{{\partial }{\mathrm{z__2}}^{{3}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F3}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right){+}\left({-}\left({\mathrm{a__2}}^{{2}}{+}\left({4}{}\mathrm{b__2}{+}{9}\right){}\mathrm{a__2}{+}{\mathrm{b__2}}^{{2}}{+}{9}{}\mathrm{b__2}{+}{14}\right){}\left(\mathrm{z__1}{-}{1}\right){}{\mathrm{z__2}}^{{2}}{+}\left(\left(\left({2}{}{c}{-}\mathrm{a__1}{-}\mathrm{b__1}{+}{2}{}\mathrm{b__2}{+}{5}\right){}\mathrm{a__2}{+}\left({2}{}{c}{-}\mathrm{a__1}{-}\mathrm{b__1}{+}{5}\right){}\mathrm{b__2}{+}{6}{}{c}{-}{3}{}\mathrm{a__1}{-}{3}{}\mathrm{b__1}{+}{11}\right){}\mathrm{z__1}{+}\left({-}{c}{-}\mathrm{b__2}{-}{2}\right){}\mathrm{a__2}{+}\left({-}{c}{-}{2}\right){}\mathrm{b__2}{-}{3}{}{c}{-}{4}\right){}\mathrm{z__2}{-}\mathrm{z__1}{}\left({c}{-}\mathrm{b__1}{+}{1}\right){}\left({c}{-}\mathrm{a__1}{+}{1}\right)\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__2}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F3}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right){+}{2}{}\left(\left({-}\left(\mathrm{z__1}{-}{1}\right){}\left(\mathrm{a__2}{+}\mathrm{b__2}{+}{2}\right){}\mathrm{z__2}{+}\left({c}{-}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{b__1}}{{2}}{+}\frac{{1}}{{2}}\right){}\mathrm{z__1}{-}\frac{{c}}{{2}}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{F3}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right){-}\frac{\mathrm{a__2}{}\mathrm{b__2}{}{\mathrm{F3}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){}\left(\mathrm{z__1}{-}{1}\right)}{{2}}\right){}\left(\mathrm{b__2}{+}{1}\right){}\left(\mathrm{a__2}{+}{1}\right)}{{\mathrm{z__2}}^{{2}}{}\left(\mathrm{z__2}{-}{1}\right){}\left(\left(\mathrm{z__1}{-}{1}\right){}\mathrm{z__2}{-}\mathrm{z__1}\right)}$ (13)

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

 > $\mathrm{DEtools}\left[\mathrm{singularities}\right]\left(\mathrm{subs}\left(\mathrm{F3}\left(\mathrm{z__1},\mathrm{z__2}\right)=\mathrm{F3}\left(\mathrm{z__2}\right),\right)\right)$
 ${\mathrm{regular}}{=}\left\{{0}{,}{1}{,}{\mathrm{\infty }}{,}\frac{\mathrm{z__1}}{\mathrm{z__1}{-}{1}}\right\}{,}{\mathrm{irregular}}{=}{\varnothing }$ (14)

You can also see a general presentation of AppellF3, organized into sections and including plots, using the FunctionAdvisor

 > $\mathrm{FunctionAdvisor}\left(\mathrm{AppellF3}\right)$

AppellF3

describe

 ${\mathrm{AppellF3}}{=}{\mathrm{Appell 2-variable hypergeometric function F3}}$

definition

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{a__2}\right)}_{{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left({c}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}$ $\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}$

classify function

 ${\mathrm{Appell}}$

symmetries

 ${{F}}_{{3}}{}\left(\mathrm{a__2}{,}\mathrm{a__1}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ ${{F}}_{{3}}{}\left(\mathrm{b__1}{,}\mathrm{a__2}{,}\mathrm{a__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}\mathrm{a__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$

plot

singularities

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left({c}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{::}\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 }\mathrm{a__2}{::}\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 }\mathrm{a__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{a__2}{::}{\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}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__1}{+}\mathrm{a__2}{<}{c}\right){\vee }\left({c}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{a__2}{::}\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){\vee }\left({c}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{a__2}{::}{\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 }\mathrm{b__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__2}{::}{\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 }\mathrm{b__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}\right){\vee }\left({c}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__2}{::}\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{b__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__2}{::}{\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 }\mathrm{b__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__2}{::}{\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 }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__2}{::}\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{a__1}{+}\mathrm{b__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{a__2}{::}{\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}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{a__2}{<}{c}\right)$

branch points

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left(\mathrm{a__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{z__1}{\in }\left[{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]\right){\vee }\left(\mathrm{a__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{z__2}{\in }\left[{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]\right)$

branch cuts

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left(\mathrm{a__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{1}{<}\mathrm{z__1}\right){\vee }\left(\mathrm{a__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{b__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{1}{<}\mathrm{z__2}\right)$

special values

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ $\mathrm{z__1}{=}{0}{\wedge }\mathrm{z__2}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ $\mathrm{a__1}{=}{0}{\wedge }\mathrm{a__2}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ $\mathrm{a__1}{=}{0}{\wedge }\mathrm{b__2}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ $\mathrm{a__2}{=}{0}{\wedge }\mathrm{b__1}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ $\mathrm{b__1}{=}{0}{\wedge }\mathrm{b__2}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{;}\mathrm{z__2}\right)$ $\mathrm{z__1}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{;}\mathrm{z__1}\right)$ $\mathrm{z__2}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{;}{1}\right){}{}_{{3}}{F}_{{2}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{,}{c}{-}\mathrm{a__1}{-}\mathrm{b__1}{;}{c}{-}\mathrm{a__1}{,}{c}{-}\mathrm{b__1}{;}\mathrm{z__2}\right)$ $\mathrm{z__1}{=}{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{;}{1}\right){}{}_{{3}}{F}_{{2}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{,}{c}{-}\mathrm{a__2}{-}\mathrm{b__2}{;}{c}{-}\mathrm{b__2}{,}{c}{-}\mathrm{a__2}{;}\mathrm{z__1}\right)$ $\mathrm{z__2}{=}{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{;}\mathrm{z__2}\right)$ $\mathrm{a__1}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{;}\mathrm{z__1}\right)$ $\mathrm{a__2}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{;}\mathrm{z__2}\right)$ $\mathrm{b__1}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{;}\mathrm{z__1}\right)$ $\mathrm{b__2}{=}{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{z__1}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{a__1}{;}\mathrm{a__1}{+}\mathrm{a__2}{;}\mathrm{z__1}\right){+}\mathrm{z__2}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{a__2}{;}\mathrm{a__1}{+}\mathrm{a__2}{;}\mathrm{z__2}\right)}{{-}\mathrm{z__1}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}}$ $\mathrm{b__1}{=}{1}{\wedge }\mathrm{b__2}{=}{1}{\wedge }{c}{=}\mathrm{a__1}{+}\mathrm{a__2}{\wedge }{-}\mathrm{z__1}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}{\ne }{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{z__1}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{b__1}{;}\mathrm{b__1}{+}\mathrm{a__2}{;}\mathrm{z__1}\right){+}\mathrm{z__2}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{a__2}{;}\mathrm{b__1}{+}\mathrm{a__2}{;}\mathrm{z__2}\right)}{{-}\mathrm{z__1}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}}$ $\mathrm{a__1}{=}{1}{\wedge }\mathrm{b__2}{=}{1}{\wedge }{c}{=}\mathrm{b__1}{+}\mathrm{a__2}{\wedge }{-}\mathrm{z__1}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}{\ne }{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{z__1}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{a__1}{;}\mathrm{a__1}{+}\mathrm{b__2}{;}\mathrm{z__1}\right){+}\mathrm{z__2}{}{}_{{2}}{F}_{{1}}{}\left({1}{,}\mathrm{b__2}{;}\mathrm{a__1}{+}\mathrm{b__2}{;}\mathrm{z__2}\right)}{{-}\mathrm{z__1}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}}$ $\mathrm{b__1}{=}{1}{\wedge }\mathrm{a__2}{=}{1}{\wedge }{c}{=}\mathrm{a__1}{+}\mathrm{b__2}{\wedge }{-}\mathrm{z__1}{}\mathrm{z__2}{+}\mathrm{z__1}{+}\mathrm{z__2}{\ne }{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__2}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{+}\mathrm{b__2}{,}\mathrm{b__1}{+}\mathrm{b__2}{;}\mathrm{a__1}{+}\mathrm{a__2}{+}\mathrm{b__1}{+}\mathrm{b__2}{;}\mathrm{z__1}\right)$ ${c}{=}\mathrm{a__1}{+}\mathrm{a__2}{+}\mathrm{b__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__2}{=}\frac{\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__2}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{+}\mathrm{b__2}{,}\mathrm{b__1}{+}\mathrm{b__2}{;}\mathrm{a__1}{+}\mathrm{a__2}{+}\mathrm{b__1}{+}\mathrm{b__2}{;}\mathrm{z__1}\right)$ ${c}{=}\mathrm{a__1}{+}\mathrm{a__2}{+}\mathrm{b__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__2}{\ne }{1}{\wedge }\mathrm{z__1}{=}\frac{\mathrm{z__2}}{{-}{1}{+}\mathrm{z__2}}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{-}\mathrm{z__1}\right)}^{{c}{-}{1}}{}{\left({1}{-}{2}{}\mathrm{z__1}\right)}^{{1}{-}\mathrm{a__2}}{}{}_{{2}}{F}_{{1}}{}\left(\frac{{c}}{{2}}{+}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{,}\frac{{c}}{{2}}{-}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{+}\frac{{1}}{{2}}{;}{c}{;}{4}{}\mathrm{z__1}{}\left({1}{-}\mathrm{z__1}\right)\right)$ $\mathrm{z__2}{=}\frac{\mathrm{z__1}}{{2}{}\mathrm{z__1}{-}{1}}{\wedge }{2}{}\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{b__1}{=}{1}{-}\mathrm{a__1}{\wedge }\mathrm{b__2}{=}{1}{-}\mathrm{a__2}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{-}\mathrm{z__1}\right)}^{{c}{-}{1}}{}{\left({1}{-}{2}{}\mathrm{z__1}\right)}^{{1}{-}\mathrm{a__2}}{}{}_{{2}}{F}_{{1}}{}\left(\frac{{c}}{{2}}{+}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{,}\frac{{c}}{{2}}{-}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{+}\frac{{1}}{{2}}{;}{c}{;}{4}{}\mathrm{z__1}{}\left({1}{-}\mathrm{z__1}\right)\right)$ $\mathrm{z__1}{=}\frac{\mathrm{z__2}}{{2}{}\mathrm{z__2}{-}{1}}{\wedge }{2}{}\mathrm{z__2}{\ne }{1}{\wedge }\mathrm{z__2}{\ne }{1}{\wedge }\mathrm{b__1}{=}{1}{-}\mathrm{a__1}{\wedge }\mathrm{b__2}{=}{1}{-}\mathrm{a__2}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{-}\mathrm{z__1}\right)}^{{c}{-}{1}}{}{\left({1}{-}{2}{}\mathrm{z__1}\right)}^{{1}{-}\mathrm{a__2}}{}{}_{{2}}{F}_{{1}}{}\left(\frac{{c}}{{2}}{+}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{,}\frac{{c}}{{2}}{-}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{+}\frac{{1}}{{2}}{;}{c}{;}{4}{}\mathrm{z__1}{}\left({1}{-}\mathrm{z__1}\right)\right)$ $\mathrm{z__2}{=}\frac{\mathrm{z__1}}{{2}{}\mathrm{z__1}{-}{1}}{\wedge }{2}{}\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{a__1}{=}{1}{-}\mathrm{b__1}{\wedge }\mathrm{b__2}{=}{1}{-}\mathrm{a__2}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left({1}{-}\mathrm{z__1}\right)}^{{c}{-}{1}}{}{\left({1}{-}{2}{}\mathrm{z__1}\right)}^{{1}{-}\mathrm{a__2}}{}{}_{{2}}{F}_{{1}}{}\left(\frac{{c}}{{2}}{+}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{,}\frac{{c}}{{2}}{-}\frac{\mathrm{a__1}}{{2}}{-}\frac{\mathrm{a__2}}{{2}}{+}\frac{{1}}{{2}}{;}{c}{;}{4}{}\mathrm{z__1}{}\left({1}{-}\mathrm{z__1}\right)\right)$ $\mathrm{z__2}{=}\frac{\mathrm{z__1}}{{2}{}\mathrm{z__1}{-}{1}}{\wedge }{2}{}\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{b__1}{=}{1}{-}\mathrm{a__1}{\wedge }\mathrm{a__2}{=}{1}{-}\mathrm{b__2}$

identities

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{+}{n}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{\left(\genfrac{}{}{0}{}{{n}}{{k}}\right){}{\left(\mathrm{a__2}\right)}_{{k}}{}{\mathrm{z__2}}^{{k}}{}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{b__2}{+}{k}{,}\mathrm{b__1}{,}{k}{+}\mathrm{a__2}{,}{k}{+}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left({c}\right)}_{{k}}}\right)$ $\mathrm{z__2}{\ne }{1}{\wedge }\left({c}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\vee }\left(\mathrm{a__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{c}{<}\mathrm{a__2}\right){\vee }{n}{\le }\left|{c}\right|\right)$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{b__1}{,}\mathrm{a__2}{,}\mathrm{a__1}{+}{n}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{\left(\genfrac{}{}{0}{}{{n}}{{k}}\right){}{\left(\mathrm{b__1}\right)}_{{k}}{}{\mathrm{z__1}}^{{k}}{}{{F}}_{{3}}{}\left(\mathrm{a__1}{+}{k}{,}\mathrm{a__2}{,}{k}{+}\mathrm{b__1}{,}\mathrm{b__2}{,}{k}{+}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left({c}\right)}_{{k}}}\right)$ $\mathrm{z__1}{\ne }{1}{\wedge }\left({c}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\vee }\left(\mathrm{b__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{c}{<}\mathrm{b__1}\right){\vee }{n}{\le }\left|{c}\right|\right)$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\mathrm{a__1}\right)}_{{n}}{}{{F}}_{{3}}{}\left(\mathrm{b__1}{,}\mathrm{a__2}{,}\mathrm{a__1}{+}{n}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left(\mathrm{a__1}{-}\mathrm{b__1}\right)}_{{n}}}{-}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{\left(\genfrac{}{}{0}{}{{n}}{{k}}\right){}{\left(\mathrm{b__1}\right)}_{{k}}{}{\left({-1}\right)}^{{k}}{}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}{k}{+}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left(\mathrm{b__1}{-}\mathrm{a__1}{-}{n}{+}{1}\right)}_{{k}}}\right)$ $\mathrm{z__1}{\ne }{1}{\wedge }\left(\left(\mathrm{a__1}{-}\mathrm{b__1}\right){::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\vee }{n}{\le }\left|\mathrm{a__1}{-}\mathrm{b__1}\right|\right){\wedge }\left(\left(\mathrm{b__1}{-}\mathrm{a__1}{-}{n}{+}{1}\right){::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\vee }\left(\mathrm{b__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }\mathrm{b__1}{-}\mathrm{a__1}{-}{n}{+}{1}{<}\mathrm{b__1}\right){\vee }{n}{\le }\left|{-}\mathrm{b__1}{+}\mathrm{a__1}{+}{n}{-}{1}\right|\right)$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{b__1}{,}\mathrm{a__2}{,}\mathrm{a__1}{+}{n}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}\frac{\mathrm{b__1}{}\mathrm{z__1}{}\left({\sum }_{{k}{=}{1}}^{{n}}{}{{F}}_{{3}}{}\left(\mathrm{a__1}{+}{k}{,}\mathrm{a__2}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}{c}{+}{1}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{{c}}$ $\mathrm{z__1}{\ne }{1}{\wedge }{c}{\ne }{0}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{-}{n}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}\mathrm{a__1}{}\mathrm{b__1}{}\mathrm{z__1}{}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{{{F}}_{{3}}{}\left(\mathrm{a__1}{+}{1}{,}\mathrm{a__2}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}{c}{+}{2}{-}{k}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\left({c}{-}{k}\right){}\left({c}{-}{k}{+}{1}\right)}\right){-}\mathrm{a__2}{}\mathrm{b__2}{}\mathrm{z__2}{}\left({\sum }_{{k}{=}{1}}^{{n}}{}\frac{{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{+}{1}{,}\mathrm{b__1}{,}\mathrm{b__2}{+}{1}{,}{c}{+}{2}{-}{k}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\left({c}{-}{k}\right){}\left({c}{-}{k}{+}{1}\right)}\right)$ $\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__2}{\ne }{1}{\wedge }{c}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{posint}}{'}\right]\right)\right)$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{z__2}{,}\frac{\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__1}}}$ ${c}{=}\mathrm{a__1}{+}\mathrm{a__2}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{,}\mathrm{a__1}{,}{c}{,}\mathrm{z__2}{,}\frac{\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{a__1}}}$ ${c}{=}\mathrm{b__1}{+}\mathrm{a__2}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{b__2}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{z__2}{,}\frac{\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__1}}}$ ${c}{=}\mathrm{a__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{b__2}{,}\mathrm{a__2}{,}\mathrm{a__1}{,}{c}{,}\mathrm{z__2}{,}\frac{\mathrm{z__1}}{{-}{1}{+}\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{a__1}}}$ ${c}{=}\mathrm{b__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\frac{\mathrm{z__2}}{{-}{1}{+}\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{b__2}}}$ ${c}{=}\mathrm{a__1}{+}\mathrm{a__2}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{b__1}{,}\mathrm{a__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\frac{\mathrm{z__2}}{{-}{1}{+}\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{b__2}}}$ ${c}{=}\mathrm{b__1}{+}\mathrm{a__2}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{,}\mathrm{a__2}{,}{c}{,}\mathrm{z__1}{,}\frac{\mathrm{z__2}}{{-}{1}{+}\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{a__2}}}$ ${c}{=}\mathrm{a__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{1}}{}\left(\mathrm{b__1}{,}\mathrm{a__1}{,}\mathrm{a__2}{,}{c}{,}\mathrm{z__1}{,}\frac{\mathrm{z__2}}{{-}{1}{+}\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{a__2}}}$ ${c}{=}\mathrm{b__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{\mathrm{z__2}{}\left({-}{1}{+}\mathrm{z__1}\right)}{\mathrm{z__1}}\right)}^{\mathrm{b__1}}{}{{F}}_{{2}}{}\left(\mathrm{b__1}{+}\mathrm{b__2}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{b__1}{+}\mathrm{b__2}{,}\mathrm{z__2}{,}{1}{-}\frac{\mathrm{z__2}{}\left({-}{1}{+}\mathrm{z__1}\right)}{\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__1}}}$ ${c}{=}\mathrm{a__1}{+}\mathrm{a__2}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{\mathrm{z__2}{}\left({-}{1}{+}\mathrm{z__1}\right)}{\mathrm{z__1}}\right)}^{\mathrm{a__1}}{}{{F}}_{{2}}{}\left(\mathrm{a__1}{+}\mathrm{b__2}{,}\mathrm{a__2}{,}\mathrm{a__1}{,}{c}{,}\mathrm{a__1}{+}\mathrm{b__2}{,}\mathrm{z__2}{,}{1}{-}\frac{\mathrm{z__2}{}\left({-}{1}{+}\mathrm{z__1}\right)}{\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{a__1}}}$ ${c}{=}\mathrm{b__1}{+}\mathrm{a__2}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{\mathrm{z__2}{}\left({-}{1}{+}\mathrm{z__1}\right)}{\mathrm{z__1}}\right)}^{\mathrm{b__1}}{}{{F}}_{{2}}{}\left(\mathrm{b__1}{+}\mathrm{a__2}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{b__1}{+}\mathrm{a__2}{,}\mathrm{z__2}{,}{1}{-}\frac{\mathrm{z__2}{}\left({-}{1}{+}\mathrm{z__1}\right)}{\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{b__1}}}$ ${c}{=}\mathrm{a__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{\mathrm{z__2}{}\left({-}{1}{+}\mathrm{z__1}\right)}{\mathrm{z__1}}\right)}^{\mathrm{a__1}}{}{{F}}_{{2}}{}\left(\mathrm{a__1}{+}\mathrm{a__2}{,}\mathrm{b__2}{,}\mathrm{a__1}{,}{c}{,}\mathrm{a__1}{+}\mathrm{a__2}{,}\mathrm{z__2}{,}{1}{-}\frac{\mathrm{z__2}{}\left({-}{1}{+}\mathrm{z__1}\right)}{\mathrm{z__1}}\right)}{{\left({1}{-}\mathrm{z__1}\right)}^{\mathrm{a__1}}}$ ${c}{=}\mathrm{b__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__1}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}}\right)}^{\mathrm{b__2}}{}{{F}}_{{2}}{}\left(\mathrm{b__1}{+}\mathrm{b__2}{,}\mathrm{a__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{b__1}{+}\mathrm{b__2}{,}\mathrm{z__1}{,}{1}{-}\frac{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{b__2}}}$ ${c}{=}\mathrm{a__1}{+}\mathrm{a__2}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}}\right)}^{\mathrm{b__2}}{}{{F}}_{{2}}{}\left(\mathrm{a__1}{+}\mathrm{b__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{a__1}{+}\mathrm{b__2}{,}\mathrm{z__1}{,}{1}{-}\frac{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{b__2}}}$ ${c}{=}\mathrm{b__1}{+}\mathrm{a__2}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}}\right)}^{\mathrm{a__2}}{}{{F}}_{{2}}{}\left(\mathrm{b__1}{+}\mathrm{a__2}{,}\mathrm{a__1}{,}\mathrm{a__2}{,}{c}{,}\mathrm{b__1}{+}\mathrm{a__2}{,}\mathrm{z__1}{,}{1}{-}\frac{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{a__2}}}$ ${c}{=}\mathrm{a__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\frac{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}}\right)}^{\mathrm{a__2}}{}{{F}}_{{2}}{}\left(\mathrm{a__1}{+}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{a__2}{,}{c}{,}\mathrm{a__1}{+}\mathrm{a__2}{,}\mathrm{z__1}{,}{1}{-}\frac{\mathrm{z__1}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}}\right)}{{\left({1}{-}\mathrm{z__2}\right)}^{\mathrm{a__2}}}$ ${c}{=}\mathrm{b__1}{+}\mathrm{b__2}{\wedge }\mathrm{z__2}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{4}}{}\left({-}\frac{{1}}{{2}}{+}{b}{+}\frac{{c}}{{2}}{,}{b}{,}{d}{,}{e}{,}{-}\frac{{2}{}\mathrm{z__2}{}\sqrt{{g}}{-}\mathrm{z__2}{}{g}{-}{4}{}\sqrt{{g}}{-}\mathrm{z__2}}{\mathrm{z__2}}{,}{g}\right){}{\left(\sqrt{{g}}{+}{1}\right)}^{{-}{1}{+}{2}{}{b}{+}{c}}{}{\left(\frac{{4}{}\sqrt{{g}}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}{}{\left(\sqrt{{g}}{+}{1}\right)}^{{2}}}\right)}^{{b}}}{{\left({1}{-}\mathrm{z__2}\right)}^{{b}}}$ $\mathrm{a__2}{=}{b}{\wedge }\mathrm{b__1}{=}\frac{{c}}{{2}}{\wedge }\mathrm{b__2}{=}\frac{{c}}{{2}}{\wedge }{e}{=}\frac{{1}}{{2}}{+}\frac{{c}}{{2}}{\wedge }\mathrm{a__1}{=}{-}\frac{{1}}{{2}}{+}\frac{{c}}{{2}}{\wedge }{d}{=}{-}\frac{{1}}{{2}}{+}{b}{+}\frac{{c}}{{2}}{\wedge }\mathrm{z__1}{=}\frac{{4}{}\sqrt{{g}}}{{\left(\sqrt{{g}}{+}{1}\right)}^{{2}}}{\wedge }{-}\frac{{2}{}\mathrm{z__2}{}\sqrt{{g}}{-}\mathrm{z__2}{}{g}{-}{4}{}\sqrt{{g}}{-}\mathrm{z__2}}{\mathrm{z__2}{}{\left(\sqrt{{g}}{+}{1}\right)}^{{2}}}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{4}}{}\left({b}{+}\frac{{1}}{{2}}{-}\frac{{c}}{{2}}{,}{b}{,}{d}{,}{e}{,}{-}\frac{{2}{}\mathrm{z__2}{}\sqrt{{g}}{-}\mathrm{z__2}{}{g}{-}{4}{}\sqrt{{g}}{-}\mathrm{z__2}}{\mathrm{z__2}}{,}{g}\right){}{\left(\sqrt{{g}}{+}{1}\right)}^{{2}{}{b}}{}{\left(\frac{{4}{}\sqrt{{g}}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}{}{\left(\sqrt{{g}}{+}{1}\right)}^{{2}}}\right)}^{{b}{+}\frac{{1}}{{2}}{-}\frac{{c}}{{2}}}}{{\left({1}{-}\mathrm{z__2}\right)}^{{b}{+}\frac{{1}}{{2}}{-}\frac{{c}}{{2}}}}$ ${d}{=}{b}{\wedge }\mathrm{b__1}{=}\frac{{c}}{{2}}{\wedge }\mathrm{b__2}{=}\frac{{c}}{{2}}{\wedge }{e}{=}\frac{{1}}{{2}}{+}\frac{{c}}{{2}}{\wedge }\mathrm{a__1}{=}{-}\frac{{1}}{{2}}{+}\frac{{c}}{{2}}{\wedge }\mathrm{a__2}{=}{b}{+}\frac{{1}}{{2}}{-}\frac{{c}}{{2}}{\wedge }\mathrm{z__1}{=}\frac{{4}{}\sqrt{{g}}}{{\left(\sqrt{{g}}{+}{1}\right)}^{{2}}}{\wedge }{-}\frac{{2}{}\mathrm{z__2}{}\sqrt{{g}}{-}\mathrm{z__2}{}{g}{-}{4}{}\sqrt{{g}}{-}\mathrm{z__2}}{\mathrm{z__2}{}{\left(\sqrt{{g}}{+}{1}\right)}^{{2}}}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{4}}{}\left({-}\frac{{1}}{{2}}{+}{b}{+}\frac{{c}}{{2}}{,}{b}{,}{d}{,}{e}{,}{f}{,}{-}\frac{{2}{}\mathrm{z__2}{}\sqrt{{f}}{-}\mathrm{z__2}{}{f}{-}{4}{}\sqrt{{f}}{-}\mathrm{z__2}}{\mathrm{z__2}}\right){}{\left(\sqrt{{f}}{+}{1}\right)}^{{-}{1}{+}{2}{}{b}{+}{c}}{}{\left(\frac{{4}{}\sqrt{{f}}{}\left({-}{1}{+}\mathrm{z__2}\right)}{\mathrm{z__2}{}{\left(\sqrt{{f}}{+}{1}\right)}^{{2}}}\right)}^{{b}}}{{\left({1}{-}\mathrm{z__2}\right)}^{{b}}}$ $\mathrm{a__2}{=}{b}{\wedge }\mathrm{b__1}{=}\frac{{c}}{{2}}{\wedge }\mathrm{b__2}{=}\frac{{c}}{{2}}{\wedge }{d}{=}\frac{{1}}{{2}}{+}\frac{{c}}{{2}}{\wedge }\mathrm{a__1}{=}{-}\frac{{1}}{{2}}{+}\frac{{c}}{{2}}{\wedge }{e}{=}{-}\frac{{1}}{{2}}{+}{b}{+}\frac{{c}}{{2}}{\wedge }\mathrm{z__1}{=}\frac{{4}{}\sqrt{{f}}}{{\left(\sqrt{{f}}{+}{1}\right)}^{{2}}}{\wedge }{-}\frac{{2}{}\mathrm{z__2}{}\sqrt{{f}}{-}\mathrm{z__2}{}{f}{-}{4}{}\sqrt{{f}}{-}\mathrm{z__2}}{\mathrm{z__2}{}{\left(\sqrt{{f}}{+}{1}\right)}^{{2}}}{\ne }{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{4}}{}\left({b}{+}\frac{{1}}{{2}}{-}\frac{{c}}{{2}}{,}{b}{,}{d}{,}{e}{,}{f}{,}{-}\frac{{2}{}\mathrm{z__2}{}\sqrt{{f}}{-}\mathrm{z__2}{}{f}{-}}{}\right)}{}$