Mathematical Functions - Maple Help

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 Mathematical Functions

Relevant developments in the MathematicalFunctions project happened for Maple 2017, regarding both the addition of the four Appell functions, representing the first ever full implementation of these functions in computational environments, as well as the addition of a new package, Evalf, for performing numerical experimentation taking advantage of sophisticated symbolic computation functionality. The Evalf package and project aims to provide a user-friendly environment to develop and work with numerical algorithms for mathematical functions.

The Four Appell Functions

The four multi-parameter Appell functions, AppellF1, AppellF2, AppellF3 and AppellF4 are doubly hypergeometric functions that include as particular cases the 2F1 hypergeometric and some cases of the MeijerG function, and with them most of the known functions of mathematical physics. These Appell functions have been popping up with increasing frequency in applications in quantum mechanics, molecular physics, and general relativity.

As in the case of the hypergeometric function, a distinction is made between the four Appell series, with restricted domain of convergence, and the four Appell functions, that coincide with the series in their domain of convergence but also extend them analytically to the whole complex plane. The Maple implementation of the Appell functions includes a thorough set of their symbolic properties, all accessible using the FunctionAdvisor, as well as numerical algorithms to evaluate the four functions over the whole complex plane, representing the first ever complete computational implementation of these functions.

To display special functions and sequences using textbook notation as shown in this page, use extended typesetting and enable the typesetting of mathematical functions

 >

Examples

 The definition of the four Appell series and the corresponding domains of convergence can be seen through the FunctionAdvisor. For example,
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left({c}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]$ (1.1.1)
 >
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left(\mathrm{c__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{c__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\left|\mathrm{z__1}\right|{+}\left|\mathrm{z__2}\right|{<}{1}\right]$ (1.1.2)
 >
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{a__2}\right)}_{{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left({c}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]$ (1.1.3)
 >
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left({b}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left(\mathrm{c__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{c__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\sqrt{\left|\mathrm{z__1}\right|}{+}\sqrt{\left|\mathrm{z__2}\right|}{<}{1}\right]$ (1.1.4)
 From these definitions, these series and the corresponding analytic extensions (Appell functions) are singular (division by zero) when the $c$ parameters entering the pochhammer functions in the denominators of these series are non-positive integers. For an analogous reason, when the $a$ and/or $b$ parameters entering the pochhammer functions in the numerators of the series are non-positive integers, the series will truncate and the Appell functions will be polynomial. Consult the FunctionAdvisor for comprehensive information on the combinations of all these conditions. For example, for AppellF1, the singular cases happen when any of the following conditions hold
 > $\mathrm{FunctionAdvisor}\left(\mathrm{singularities},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }{a}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }{a}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }{a}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }{a}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }{a}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }{a}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }{a}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right)\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }{a}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right)\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }{a}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right)\right]$ (1.1.5)
 By requesting the sum form of the Appell functions, besides their double power series definition, we also see the particular form the four series take when one of the summations is performed and the result expressed in terms of 2F1 hypergeometric functions. For example, for AppellF3,
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{m}}{}{\left(\mathrm{a__2}\right)}_{{n}}{}{\left(\mathrm{b__1}\right)}_{{m}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left({c}\right)}_{{m}{+}{n}}{}{m}{!}{}{n}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{k}}{}{\left(\mathrm{b__1}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{+}{k}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__2}\right)}_{{k}}{}{\left(\mathrm{b__2}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{+}{k}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__2}\right|{<}{1}\right]$ (1.1.6)
 So, for AppellF3 (and also for AppellF1, but not for AppellF2 nor AppellF4) 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.

In the literature, the Appell series are analytically extended by integral representations in terms of Eulerian double integrals. With the exception of AppellF4, one of the two iterated integrals can always be computed resulting in a single integral with hypergeometric integrand. For example, for AppellF2

 >
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__1}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{{u}}^{\mathrm{b__1}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}{-}\frac{\mathrm{z__2}}{{u}{}\mathrm{z__1}{-}{1}}\right)}{{\left({1}{-}{u}\right)}^{{-}\mathrm{c__1}{+}\mathrm{b__1}{+}{1}}{}{\left({-}{u}{}\mathrm{z__1}{+}{1}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__1}{-}\mathrm{b__1}\right)}{,}\mathrm{z__1}{\ne }{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__1}{+}\mathrm{b__1}\right)\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{{u}}^{\mathrm{b__2}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}{-}\frac{\mathrm{z__1}}{{u}{}\mathrm{z__2}{-}{1}}\right)}{{\left({1}{-}{u}\right)}^{{1}{+}\mathrm{b__2}{-}\mathrm{c__2}}{}{\left({-}{u}{}\mathrm{z__2}{+}{1}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}\mathrm{b__2}\right)}{,}\mathrm{z__2}{\ne }{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__2}{+}\mathrm{b__2}\right)\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}\left({{\int }}_{{0}}^{{1}}{{\int }}_{{0}}^{{1}}\frac{{{u}}^{\mathrm{b__1}{-}{1}}{}{{v}}^{\mathrm{b__2}{-}{1}}}{{\left({1}{-}{u}\right)}^{{-}\mathrm{c__1}{+}\mathrm{b__1}{+}{1}}{}{\left({1}{-}{v}\right)}^{{1}{+}\mathrm{b__2}{-}\mathrm{c__2}}{}{\left({-}{u}{}\mathrm{z__1}{-}{v}{}\mathrm{z__2}{+}{1}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{v}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__1}{-}\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}\mathrm{b__2}\right)}{,}{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__1}{+}\mathrm{b__1}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__2}{+}\mathrm{b__2}\right)\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{\int }}_{{0}}^{{\mathrm{\infty }}}\frac{{{u}}^{{a}{-}{1}}{}{}_{{1}}{F}_{{1}}{}\left(\mathrm{b__1}{;}\mathrm{c__1}{;}{u}{}\mathrm{z__1}\right){}{}_{{1}}{F}_{{1}}{}\left(\mathrm{b__2}{;}\mathrm{c__2}{;}{u}{}\mathrm{z__2}\right)}{{{ⅇ}}^{{u}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}}{{\mathrm{\Gamma }}{}\left({a}\right)}{,}{\mathrm{\Re }}{}\left(\mathrm{z__1}{+}\mathrm{z__2}\right){<}{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left({a}\right)\right]$ (1.1.7)
 For the purpose of numerically evaluating the four Appell functions over the whole complex plane, instead of numerically evaluating the integral representations, it is simpler, when possible, to evaluate the function using identities. For example, with the exception of AppellF3, the Appell functions admit identities analogous to Euler identities for the hypergeometric function. These Euler-type identities, as well as contiguity identities for the four Appell functions, are visible using the FunctionAdvisor with the option identities, or directly from the function. For AppellF4, for instance, provided that none of $a$, $b$, $a-b$, ${c}_{2}-a$ is a non-positive integer,
 > ${F}_{4}\left(a,b,\mathrm{c__1},\mathrm{c__2},\mathrm{z__1},\mathrm{z__2}\right)={\left({\mathrm{AppellF4}:-\mathrm{Transformations}}_{"Euler"}\right)}_{1}\left(a,b,\mathrm{c__1},\mathrm{c__2},\mathrm{z__1},\mathrm{z__2}\right)$
 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({b}{-}{a}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{a}}{}{{F}}_{{4}}{}\left({a}{,}{a}{-}\mathrm{c__2}{+}{1}{,}{a}{-}{b}{+}{1}{,}\mathrm{c__1}{,}\frac{{1}}{\mathrm{z__2}}{,}\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{a}\right){}{\mathrm{\Gamma }}{}\left({b}\right)}{+}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({a}{-}{b}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{b}}{}{{F}}_{{4}}{}\left({b}{,}{1}{+}{b}{-}\mathrm{c__2}{,}{b}{-}{a}{+}{1}{,}\mathrm{c__1}{,}\frac{{1}}{\mathrm{z__2}}{,}\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{b}\right){}{\mathrm{\Gamma }}{}\left({a}\right)}$ (1.1.8)
 and this identity can be used to evaluate AppellF4 at ${z}_{1}=1$ over the whole complex plane since, in that case, the two variables of the Appell Functions on right-hand side become equal, and that is a special case of AppellF4, expressible in terms of hypergeometric 4F3 functions
 > $\genfrac{}{}{0}{}{\phantom{\mathrm{z__1}=1}}{}|\genfrac{}{}{0}{}{\phantom{}}{\mathrm{z__1}=1}$
 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}{1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({b}{-}{a}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{a}}{}{}_{{4}}{F}_{{3}}{}\left({a}{,}{a}{-}\mathrm{c__2}{+}{1}{,}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}{+}\frac{\mathrm{c__1}}{{2}}{,}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}{+}\frac{\mathrm{c__1}}{{2}}{;}\mathrm{c__1}{,}{a}{-}{b}{+}{1}{,}{a}{-}{b}{+}\mathrm{c__1}{;}\frac{{4}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{a}\right){}{\mathrm{\Gamma }}{}\left({b}\right)}{+}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({a}{-}{b}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{b}}{}{}_{{4}}{F}_{{3}}{}\left({b}{,}{1}{+}{b}{-}\mathrm{c__2}{,}\frac{{b}}{{2}}{-}\frac{{a}}{{2}}{+}\frac{\mathrm{c__1}}{{2}}{,}\frac{{b}}{{2}}{-}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{+}\frac{\mathrm{c__1}}{{2}}{;}\mathrm{c__1}{,}{b}{-}{a}{+}{1}{,}{b}{-}{a}{+}\mathrm{c__1}{;}\frac{{4}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{b}\right){}{\mathrm{\Gamma }}{}\left({a}\right)}$ (1.1.9)

A plot of the AppellF2 function for some values of its parameters

 >
 ${\mathrm{F2}}{≔}{{F}}_{{2}}{}\left({{ⅇ}}^{{I}{}{z}}{,}\frac{{I}}{{2}}{,}\frac{{3}}{{7}}{-}\frac{{I}}{{4}}{,}{4}{,}\frac{{5}}{{7}}{+}{6}{}{I}{,}{6}{,}{z}\right)$ (1.1.10)
 > $\mathrm{plot}\left(\left[\mathrm{Re},\mathrm{Im}\right]\left(\mathrm{F2}\right),z=-1..1\right)$
 A thorough set with the main symbolic properties of any of the four Appell functions, for instance for AppellF3, can be seen via
 > $\mathrm{FunctionAdvisor}\left(\mathrm{AppellF3}\right)$

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}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right)\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right)\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{a__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{a__2}{+}\mathrm{b__1}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{a__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{a__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__2}{+}\mathrm{b__1}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}{\wedge }\mathrm{a__2}{+}\mathrm{b__1}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{+}\mathrm{b__2}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{a__2}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__1}{::}{{ℤ}}^{\left({0}{,}{-}\right)}{\wedge }\mathrm{b__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{a__1}{+}\mathrm{a__2}{<}{c}{\wedge }\mathrm{a__2}{+}\mathrm{b__1}{<}{c}\right)$

branch points

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left(\mathrm{a__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{z__1}{\in }\left[{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]\right){\vee }\left(\mathrm{a__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{z__2}{\in }\left[{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]\right)$

branch cuts

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left(\mathrm{a__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__1}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }{1}{<}\mathrm{z__1}\right){\vee }\left(\mathrm{a__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }\mathrm{b__2}{::}\left({¬}{{ℤ}}^{\left({0}{,}{-}\right)}\right){\wedge }{1}{<}\mathrm{z__2}\right)$

special values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

identities

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

sum form

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

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{k}}{}{\left(\mathrm{b__1}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{+}{k}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}$ $\left|\mathrm{z__1}\right|{<}{1}$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__2}\right)}_{{k}}{}{\left(\mathrm{b__2}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{+}{k}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}$ $\left|\mathrm{z__2}\right|{<}{1}$

series

 ${\mathrm{series}}{}\left({{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\mathrm{z__1}{,}{4}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{;}\mathrm{z__2}\right){+}\frac{\mathrm{a__1}{}\mathrm{b__1}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{+}{1}{;}\mathrm{z__2}\right)}{{c}}{}\mathrm{z__1}{+}\frac{{1}}{{2}}{}\frac{\mathrm{a__1}{}\mathrm{b__1}{}\left(\mathrm{a__1}{+}{1}\right){}\left(\mathrm{b__1}{+}{1}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{+}{2}{;}\mathrm{z__2}\right)}{{c}{}\left({c}{+}{1}\right)}{}{\mathrm{z__1}}^{{2}}{+}\frac{{1}}{{6}}{}\frac{\mathrm{a__1}{}\mathrm{b__1}{}\left(\mathrm{a__1}{+}{1}\right){}\left(\mathrm{b__1}{+}{1}\right){}\left(\mathrm{a__1}{+}{2}\right){}\left(\mathrm{b__1}{+}{2}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{+}{3}{;}\mathrm{z__2}\right)}{{c}{}\left({c}{+}{1}\right){}\left({c}{+}{2}\right)}{}{\mathrm{z__1}}^{{3}}{+}{O}{}\left({\mathrm{z__1}}^{{4}}\right)$

 ${\mathrm{series}}{}\left({{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\mathrm{z__2}{,}{4}\right){=}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{;}\mathrm{z__1}\right){+}\frac{\mathrm{a__2}{}\mathrm{b__2}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{+}{1}{;}\mathrm{z__1}\right)}{{c}}{}\mathrm{z__2}{+}\frac{{1}}{{2}}{}\frac{\mathrm{a__2}{}\mathrm{b__2}{}\left(\mathrm{a__2}{+}{1}\right){}\left(\mathrm{b__2}{+}{1}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{+}{2}{;}\mathrm{z__1}\right)}{{c}{}\left({c}{+}{1}\right)}{}{\mathrm{z__2}}^{{2}}{+}\frac{{1}}{{6}}{}\frac{\mathrm{a__2}{}\mathrm{b__2}{}\left(\mathrm{a__2}{+}{1}\right){}\left(\mathrm{b__2}{+}{1}\right){}\left(\mathrm{a__2}{+}{2}\right){}\left(\mathrm{b__2}{+}{2}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{+}{3}{;}\mathrm{z__1}\right)}{{c}{}\left({c}{+}{1}\right){}\left({c}{+}{2}\right)}{}{\mathrm{z__2}}^{{3}}{+}{O}{}\left({\mathrm{z__2}}^{{4}}\right)$

integral form

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{\left({1}{-}{u}\right)}^{{-}{1}{+}\mathrm{b__1}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{-}\mathrm{b__1}{;}\mathrm{z__2}{}{u}\right)}{{{u}}^{{-}{c}{+}\mathrm{b__1}{+}{1}}{}{\left({1}{+}\mathrm{z__1}{}\left({u}{-}{1}\right)\right)}^{\mathrm{a__1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left({c}{-}\mathrm{b__1}\right)}$ ${0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({c}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}{c}{+}\mathrm{b__1}\right)$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{\left({1}{-}{u}\right)}^{\mathrm{b__2}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{-}\mathrm{b__2}{;}{u}{}\mathrm{z__1}\right)}{{{u}}^{{-}{c}{+}\mathrm{b__2}{+}{1}}{}{\left({1}{+}\left({u}{-}{1}\right){}\mathrm{z__2}\right)}^{\mathrm{a__2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left({c}{-}\mathrm{b__2}\right)}$ ${0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({c}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}{c}{+}\mathrm{b__2}\right)$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}{{u}}^{{\mathrm{\rho }}{-}{1}}{}{\left({1}{-}{u}\right)}^{{c}{-}{\mathrm{\rho }}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{\mathrm{\rho }}{;}{u}{}\mathrm{z__1}\right){}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{-}{\mathrm{\rho }}{;}{-}\left({u}{-}{1}\right){}\mathrm{z__2}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left({\mathrm{\rho }}\right){}{\mathrm{\Gamma }}{}\left({c}{-}{\mathrm{\rho }}\right)}$ ${0}{<}{\mathrm{\Re }}{}\left({c}\right)$

 ${{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}{{\int }}_{{0}}^{{1}{-}{v}}\frac{{{u}}^{{-}{1}{+}\mathrm{b__1}}{}{{v}}^{\mathrm{b__2}{-}{1}}}{{\left({1}{-}{u}{-}{v}\right)}^{{-}{c}{+}\mathrm{b__1}{+}\mathrm{b__2}{+}{1}}{}{\left({-}{u}{}\mathrm{z__1}{+}{1}\right)}^{\mathrm{a__1}}{}{\left({-}{v}{}\mathrm{z__2}{+}{1}\right)}^{\mathrm{a__2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{v}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{b__2}\right){}{\mathrm{\Gamma }}{}\left({c}{-}\mathrm{b__1}{-}\mathrm{b__2}\right)}$ ${0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__2}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}{c}{+}\mathrm{b__1}{+}\mathrm{b__2}\right)$

differentiation rule

 $\frac{{\partial }}{{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{a__1}{}\mathrm{b__1}{}{{F}}_{{3}}{}\left(\mathrm{a__1}{+}{1}{,}\mathrm{a__2}{,}\mathrm{b__1}{+}{1}{,}\mathrm{b__2}{,}{c}{+}{1}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{c}}$

 $\frac{{{\partial }}^{{n}}}{{\partial }{\mathrm{z__1}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\mathrm{a__1}\right)}_{{n}}{}{\left(\mathrm{b__1}\right)}_{{n}}{}{{F}}_{{3}}{}\left({n}{+}\mathrm{a__1}{,}\mathrm{a__2}{,}{n}{+}\mathrm{b__1}{,}\mathrm{b__2}{,}{n}{+}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left({c}\right)}_{{n}}}$

 $\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{a__2}{}\mathrm{b__2}{}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{+}{1}{,}\mathrm{b__1}{,}\mathrm{b__2}{+}{1}{,}{c}{+}{1}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{c}}$

 $\frac{{{\partial }}^{{n}}}{{\partial }{\mathrm{z__2}}^{{n}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\left(\mathrm{a__2}\right)}_{{n}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}{n}{+}\mathrm{a__2}{,}\mathrm{b__1}{,}{n}{+}\mathrm{b__2}{,}{n}{+}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\left({c}\right)}_{{n}}}$

DE

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

 $\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__1}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{z__2}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }\mathrm{z__2}{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{+}\frac{\left(\left({-}\mathrm{a__1}{-}\mathrm{b__1}{-}{1}\right){}\mathrm{z__1}{+}{c}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{-}\frac{\mathrm{a__1}{}\mathrm{b__1}{}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}$ $\frac{{{\partial }}^{{2}}}{{\partial }\mathrm{z__2}{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{\mathrm{z__2}{}\left(\mathrm{z__2}{-}{1}\right){}\left(\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__2}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}}{+}\frac{\left(\left(\mathrm{a__2}{+}\mathrm{b__2}{+}{1}\right){}\mathrm{z__2}{-}{c}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}}{+}\frac{\mathrm{a__2}{}\mathrm{b__2}{}{f}{}\left(\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\mathrm{z__1}}$

The Evalf Package

Evalf is both a command and a package of commands for the numerical evaluation of mathematical expressions and functions, numerical experimentation, and fast development of numerical algorithms, taking advantage of the advanced symbolic capabilities of the Maple computer algebra system.

As an environment for working with special functions, Evalf helps developing/implementing the typical approaches used in the literature and comparing their performances. This kind of environment is increasingly relevant nowadays, when rather complicated mathematical expressions and advanced special functions, as for instance is the case of the Heun and Appell functions, appear more and more in the modeling of problems in science.

Examples

 >
 $\left\{{\mathrm{Add}}{,}{\mathrm{Evalb}}{,}{\mathrm{Zoom}}{,}{\mathrm{QuadrantNumbers}}{,}{\mathrm{Singularities}}{,}{\mathrm{GenerateRecurrence}}{,}{\mathrm{PairwiseSummation}}\right\}$ (2.1.1)

Consider the following AppellF4 function

 > $\mathrm{F4}≔\mathrm{AppellF4}\left(1,2,3,4,5,z\right)$
 ${\mathrm{F4}}{≔}{{F}}_{{4}}{}\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{z}\right)$ (2.1.2)

This function satisfies a linear differential equation whose singularities, which depend on the function's parameters, are relevant in the context of numerically evaluating the function. To see the location of these singularities you can construct the linear ODE behind F4 using PDEtools:-dpolyform) and use the DEtools:-singularities command, or directly use Evalf:-Singularities

 > $S≔\mathrm{Singularities}\left(\right)$