AppellF4 - Maple Help

AppellF4

The AppellF4 function

 Calling Sequence AppellF4($a,b,{c}_{1},{c}_{2},{z}_{1},{z}_{2}$)

Parameters

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

Description

 • As is the case of all the four multi-parameter Appell functions, AppellF4, 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, AppellF4 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 AppellF4 series and the corresponding domain of convergence can be seen through the FunctionAdvisor
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{AppellF4}\right)$
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left({b}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left(\mathrm{c__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{c__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\sqrt{\left|\mathrm{z__1}\right|}{+}\sqrt{\left|\mathrm{z__2}\right|}{<}{1}\right]$ (1)
 A distinction is made between the AppellF4 doubly hypergeometric series, with the restricted domain of convergence shown above, and the AppellF4 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 AppellF4 variables subscripted with the numbers 1 and 2, the function remains the same; hence
 > $\mathrm{FunctionAdvisor}\left(\mathrm{symmetries},\mathrm{AppellF4}\right)$
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__2}{,}\mathrm{c__1}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}{{F}}_{{4}}{}\left({b}{,}{a}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right]$ (2)
 Note the existence of another symmetry, also visible in the double sum definition.
 From the series' definition, AppellF4 is singular (division by zero) when the ${c}_{1}$ and/or ${c}_{2}$ parameters entering the pochhammer functions in the denominator of the series are non-positive integers, because these pochhammer functions will be equal to zero when the summation index of the series is bigger than the absolute value of the corresponding ${c}_{1}$ or ${c}_{2}$ parameter.
 For an analogous reason, when the $a$ and/or both ${b}_{1}$ and ${b}_{2}$ parameters entering the pochhammer functions in the numerator of the series are non-positive integers, the series will truncate and AppellF4 will be polynomial. As is the case of the hypergeometric function, when the pochhammers in both the numerator and the denominator have non-positive integer arguments, AppellF4 is polynomial if the absolute value of the non-positive integers in the pochhammers of the numerator are smaller than or equal to the absolute value of the non-positive integer (parameters ${c}_{1},{c}_{2}$) in the pochhammers in the denominator, and singular otherwise. Consult the FunctionAdvisor for comprehensive information on the combinations of all these conditions. For example, the singular cases happen when any of the following conditions hold
 > $\mathrm{FunctionAdvisor}\left(\mathrm{singularities},\mathrm{AppellF4}\right)$
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\left(\mathrm{c__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{b}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left(\mathrm{c__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{b}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{a}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{b}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{b}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{b}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{<}\mathrm{c__1}{\wedge }{b}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{b}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left(\mathrm{c__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{b}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{a}{<}\mathrm{c__2}\right){\vee }\left(\mathrm{c__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{b}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{b}{<}\mathrm{c__2}\right){\vee }\left(\mathrm{c__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{b}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{<}\mathrm{c__2}{\wedge }{b}{<}\mathrm{c__2}\right)\right]$ (3)
 The AppellF4 series is analytically extended to the AppellF4 function defined over the whole complex plane using identities and mainly by integral representations in terms of Eulerian integrals:
 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral_form},\mathrm{AppellF4}\right)$
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\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}}^{{b}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left(\frac{{a}}{{2}}{,}\frac{{1}}{{2}}{+}\frac{{a}}{{2}}{;}\mathrm{c__1}{;}\frac{{4}{}{{u}}^{{2}}{}\mathrm{z__1}{}\mathrm{z__2}}{{\left({-}{1}{+}\left(\mathrm{z__1}{+}\mathrm{z__2}\right){}{u}\right)}^{{2}}}\right)}{{\left({1}{-}{u}\right)}^{{b}{-}\mathrm{c__1}{+}{1}}{}{\left({1}{+}\left({-}\mathrm{z__1}{-}\mathrm{z__2}\right){}{u}\right)}^{{a}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left({b}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__1}{-}{b}\right)}{,}\left(\sqrt{\left|\mathrm{z__1}\right|}{+}\sqrt{\left|\mathrm{z__2}\right|}{<}{1}{\wedge }\mathrm{c__1}{=}\mathrm{c__2}{\wedge }{0}{<}{\mathrm{\Re }}{}\left({b}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{c__1}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__1}{+}{b}\right)\right){\vee }\left(\sqrt{\left|\mathrm{z__1}\right|}{+}\sqrt{\left|\mathrm{z__2}\right|}{<}{1}{\wedge }\mathrm{c__1}{=}\mathrm{c__2}{\wedge }{0}{<}{\mathrm{\Re }}{}\left({b}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left(\mathrm{c__2}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__2}{+}{b}\right)\right)\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{\int }}_{{0}}^{{\mathrm{\infty }}}\frac{{{u}}^{{2}{}{a}{-}{1}}{}{}_{{0}}{F}_{{1}}{}\left({;}\mathrm{c__1}{;}\frac{\mathrm{z__1}{}{{u}}^{{2}}}{{4}}\right){}{}_{{0}}{F}_{{1}}{}\left({;}\mathrm{c__2}{;}\frac{\mathrm{z__2}{}{{u}}^{{2}}}{{4}}\right)}{{{ⅇ}}^{{u}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}}{{\mathrm{\Gamma }}{}\left({2}{}{a}\right)}{,}\sqrt{\left|\mathrm{z__1}\right|}{+}\sqrt{\left|\mathrm{z__2}\right|}{<}{1}{\wedge }{b}{=}{a}{+}\frac{{1}}{{2}}{\wedge }{0}{<}{\mathrm{\Re }}{}\left({a}\right){\wedge }{\mathrm{\Re }}{}\left(\sqrt{\mathrm{z__1}}{+}\sqrt{\mathrm{z__2}}\right){<}{1}{\wedge }{\mathrm{\Re }}{}\left(\sqrt{\mathrm{z__1}}{-}\sqrt{\mathrm{z__2}}\right){<}{1}{\wedge }{-}{\mathrm{\Re }}{}\left(\sqrt{\mathrm{z__1}}{-}\sqrt{\mathrm{z__2}}\right){<}{1}{\wedge }{-}{\mathrm{\Re }}{}\left(\sqrt{\mathrm{z__1}}{+}\sqrt{\mathrm{z__2}}\right){<}{1}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\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}}^{{a}{-}{1}}{}{{v}}^{{b}{-}{1}}}{{\left({1}{-}{u}\right)}^{{-}\mathrm{c__1}{+}{a}{+}{1}}{}{\left({1}{-}{v}\right)}^{{-}\mathrm{c__2}{+}{b}{+}{1}}{}\left({1}{-}\frac{{u}{}{\left({1}{+}\frac{{u}{}\left({-}{\mathrm{α1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){+}\mathrm{z__2}\right)}{{\mathrm{α1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)}{-}{v}{}{\mathrm{α1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}^{{-}\mathrm{c__1}{-}\mathrm{c__2}{+}{a}{+}{1}{+}{b}}}{{\left(\frac{{\mathrm{α1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){-}\mathrm{z__2}}{{\mathrm{α1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)}\right)}^{{-}\mathrm{c__1}{-}\mathrm{c__2}{+}{a}{+}{1}}{}{\left({1}{-}{v}{}{\mathrm{α1}}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}^{{-}\mathrm{c__1}{-}\mathrm{c__2}{+}{b}{+}{1}}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{v}\right)}{{\mathrm{\Gamma }}{}\left({a}\right){}{\mathrm{\Gamma }}{}\left({b}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__1}{-}{a}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{b}\right)}{,}\sqrt{\left|\mathrm{z__1}\right|}{+}\sqrt{\left|\mathrm{z__2}\right|}{<}{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left({a}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({b}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__1}{+}{a}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__2}{+}{b}\right){\wedge }{{\mathrm{\alpha }}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)}^{{2}}{+}\left(\mathrm{z__1}{-}\mathrm{z__2}{-}{1}\right){}{\mathrm{\alpha }}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){+}\mathrm{z__2}{=}{0}\right]{,}\left[{{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(\mathrm{c__1}\right){}\left({{\int }}_{{0}}^{{1}}{{\int }}_{{0}}^{{1}}\frac{{{u}}^{{a}{-}{1}}{}{{v}}^{{b}{-}{1}}}{{\left({1}{-}{u}\right)}^{{-}\mathrm{c__2}{+}{a}{+}{1}}{}{\left({1}{-}{v}\right)}^{{b}{-}\mathrm{c__1}{+}{1}}{}\left({1}{-}\frac{{u}{}{\left({1}{+}\frac{{u}{}\left({-}{\mathrm{α1}}{}\left(\mathrm{z__2}{,}\mathrm{z__1}\right){+}\mathrm{z__1}\right)}{{\mathrm{α1}}{}\left(\mathrm{z__2}{,}\mathrm{z__1}\right)}{-}{v}{}{\mathrm{α1}}{}\left(\mathrm{z__2}{,}\mathrm{z__1}\right)\right)}^{{-}\mathrm{c__1}{-}\mathrm{c__2}{+}{a}{+}{1}{+}{b}}}{{\left(\frac{{\mathrm{α1}}{}\left(\mathrm{z__2}{,}\mathrm{z__1}\right){-}\mathrm{z__1}}{{\mathrm{α1}}{}\left(\mathrm{z__2}{,}\mathrm{z__1}\right)}\right)}^{{-}\mathrm{c__1}{-}\mathrm{c__2}{+}{a}{+}{1}}{}{\left({1}{-}{v}{}{\mathrm{α1}}{}\left(\mathrm{z__2}{,}\mathrm{z__1}\right)\right)}^{{-}\mathrm{c__1}{-}\mathrm{c__2}{+}{b}{+}{1}}}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{v}\right)}{{\mathrm{\Gamma }}{}\left({a}\right){}{\mathrm{\Gamma }}{}\left({b}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{a}\right){}{\mathrm{\Gamma }}{}\left(\mathrm{c__1}{-}{b}\right)}{,}\sqrt{\left|\mathrm{z__1}\right|}{+}\sqrt{\left|\mathrm{z__2}\right|}{<}{1}{\wedge }{0}{<}{\mathrm{\Re }}{}\left({a}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({b}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__2}{+}{a}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}\mathrm{c__1}{+}{b}\right){\wedge }{{\mathrm{\alpha }}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right)}^{{2}}{+}\left(\mathrm{z__1}{-}\mathrm{z__2}{-}{1}\right){}{\mathrm{\alpha }}{}\left(\mathrm{z__1}{,}\mathrm{z__2}\right){+}\mathrm{z__2}{=}{0}\right]$ (4)
 AppellF4 is the only one among the four Appell functions that has no single integral representation in the general case (all of its parameters arbitrary). These integral representations are also the starting point for the derivation of many of the identities known for AppellF4.
 AppellF4 also satisfies a linear system of partial differential equations of second order
 > $\mathrm{FunctionAdvisor}\left(\mathrm{DE},\mathrm{AppellF4}\right)$
 $\left[{f}{}\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){,}\left[\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__1}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{-}\frac{{2}{}\mathrm{z__2}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }\mathrm{z__1}{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{-}{1}}{-}\frac{{\mathrm{z__2}}^{{2}}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__2}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{+}\frac{\left(\left({-}{a}{-}{b}{-}{1}\right){}\mathrm{z__1}{+}\mathrm{c__1}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{-}\frac{\mathrm{z__2}{}\left({a}{+}{b}{+}{1}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{z__1}{}\left(\mathrm{z__1}{-}{1}\right)}{-}\frac{{a}{}{b}{}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\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({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{-}\frac{\mathrm{z__1}{}\left(\frac{{{\partial }}^{{2}}}{{\partial }{\mathrm{z__1}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{{2}{}\mathrm{z__2}}{-}\frac{\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({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{{2}{}\mathrm{z__1}}{-}\frac{\left({a}{+}{b}{+}{1}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__1}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{{2}{}\mathrm{z__2}}{+}\frac{\left(\left({-}{a}{-}{b}{-}{1}\right){}\mathrm{z__2}{+}\mathrm{c__2}\right){}\left(\frac{{\partial }}{{\partial }\mathrm{z__2}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{{2}{}\mathrm{z__1}{}\mathrm{z__2}}{-}\frac{{a}{}{b}{}{f}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{2}{}\mathrm{z__1}{}\mathrm{z__2}}\right]\right]$ (5)

Examples

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

The conditions for both the singular and the polynomial cases can also be seen from the AppellF4. For example, the fourteen polynomial cases of AppellF4 are

 > $\mathrm{AppellF4}:-\mathrm{SpecialValues}:-\mathrm{Polynomial}\left(\right)$
 ${8}{,}\left({a}{,}{b}{,}{\mathrm{c1}}{,}{\mathrm{c2}}{,}{\mathrm{z1}}{,}{\mathrm{z2}}\right){↦}\left[\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{b}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c1}}{\le }{a}\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{\le }{a}\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{\le }{a}{,}{\mathrm{c2}}{\le }{a}\right]{,}\left[{b}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c1}}{\le }{b}\right]{,}\left[{b}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{\le }{b}\right]{,}\left[{b}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c2}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{c1}}{\le }{b}{,}{\mathrm{c2}}{\le }{b}\right]\right]$ (6)

Likewise, the conditions for the singular cases of AppellF4 can be seen either using the FunctionAdvisor or entering AppellF4:-Singularities(), so with no arguments.

For particular values of its parameters, AppellF4 is related to the hypergeometric function. These hypergeometric cases are returned automatically. For example, for ${c}_{1}={c}_{2},{z}_{1}=-{z}_{2}$,

 > $\left(\mathrm{%AppellF4}=\mathrm{AppellF4}\right)\left(a,b,\mathrm{c__2},\mathrm{c__2},-\mathrm{z__2},\mathrm{z__2}\right)$
 ${\mathrm{%AppellF4}}{}\left({a}{,}{b}{,}\mathit{c__2}{,}\mathit{c__2}{,}{-}\mathit{z__2}{,}\mathit{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{,}\frac{{b}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}{;}\mathrm{c__2}{,}\frac{\mathrm{c__2}}{{2}}{,}\frac{\mathrm{c__2}}{{2}}{+}\frac{{1}}{{2}}{;}{-}{4}{}{\mathrm{z__2}}^{{2}}\right)$ (7)

To see all the hypergeometric cases, enter

 > $\mathrm{FunctionAdvisor}\left(\mathrm{specialize},\mathrm{AppellF4},\mathrm{hypergeom}\right)$
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}{b}{;}\mathrm{c__2}{;}\mathrm{z__2}\right){,}\mathrm{z__1}{=}{0}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}{b}{;}\mathrm{c__1}{;}\mathrm{z__1}\right){,}\mathrm{z__2}{=}{0}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{}_{{2}}{F}_{{1}}{}\left({a}{,}{a}{+}\frac{{1}}{{2}}{;}\mathrm{c__1}{;}\frac{\mathrm{z__1}}{{\left({1}{+}\sqrt{\mathrm{z__2}}\right)}^{{2}}}\right)}{{2}{}{\left({1}{+}\sqrt{\mathrm{z__2}}\right)}^{{2}{}{a}}}{+}\frac{{}_{{2}}{F}_{{1}}{}\left({a}{,}{a}{+}\frac{{1}}{{2}}{;}\mathrm{c__1}{;}\frac{\mathrm{z__1}}{{\left(\sqrt{\mathrm{z__2}}{-}{1}\right)}^{{2}}}\right)}{{2}{}{\left({1}{-}\sqrt{\mathrm{z__2}}\right)}^{{2}{}{a}}}{,}{b}{=}{a}{+}\frac{{1}}{{2}}{\wedge }\mathrm{c__2}{=}\frac{{1}}{{2}}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{}_{{2}}{F}_{{1}}{}\left({a}{,}{a}{+}\frac{{1}}{{2}}{;}\mathrm{c__2}{;}\frac{\mathrm{z__2}}{{\left({1}{+}\sqrt{\mathrm{z__1}}\right)}^{{2}}}\right)}{{2}{}{\left({1}{+}\sqrt{\mathrm{z__1}}\right)}^{{2}{}{a}}}{+}\frac{{}_{{2}}{F}_{{1}}{}\left({a}{,}{a}{+}\frac{{1}}{{2}}{;}\mathrm{c__2}{;}\frac{\mathrm{z__2}}{{\left(\sqrt{\mathrm{z__1}}{-}{1}\right)}^{{2}}}\right)}{{2}{}{\left({1}{-}\sqrt{\mathrm{z__1}}\right)}^{{2}{}{a}}}{,}{b}{=}{a}{+}\frac{{1}}{{2}}{\wedge }\mathrm{c__1}{=}\frac{{1}}{{2}}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{}_{{2}}{F}_{{1}}{}\left({b}{,}{b}{+}\frac{{1}}{{2}}{;}\mathrm{c__1}{;}\frac{\mathrm{z__1}}{{\left({1}{+}\sqrt{\mathrm{z__2}}\right)}^{{2}}}\right)}{{2}{}{\left({1}{+}\sqrt{\mathrm{z__2}}\right)}^{{2}{}{b}}}{+}\frac{{}_{{2}}{F}_{{1}}{}\left({b}{,}{b}{+}\frac{{1}}{{2}}{;}\mathrm{c__1}{;}\frac{\mathrm{z__1}}{{\left(\sqrt{\mathrm{z__2}}{-}{1}\right)}^{{2}}}\right)}{{2}{}{\left({1}{-}\sqrt{\mathrm{z__2}}\right)}^{{2}{}{b}}}{,}{a}{=}{b}{+}\frac{{1}}{{2}}{\wedge }\mathrm{c__2}{=}\frac{{1}}{{2}}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{}_{{2}}{F}_{{1}}{}\left({b}{,}{b}{+}\frac{{1}}{{2}}{;}\mathrm{c__2}{;}\frac{\mathrm{z__2}}{{\left({1}{+}\sqrt{\mathrm{z__1}}\right)}^{{2}}}\right)}{{2}{}{\left({1}{+}\sqrt{\mathrm{z__1}}\right)}^{{2}{}{b}}}{+}\frac{{}_{{2}}{F}_{{1}}{}\left({b}{,}{b}{+}\frac{{1}}{{2}}{;}\mathrm{c__2}{;}\frac{\mathrm{z__2}}{{\left(\sqrt{\mathrm{z__1}}{-}{1}\right)}^{{2}}}\right)}{{2}{}{\left({1}{-}\sqrt{\mathrm{z__1}}\right)}^{{2}{}{b}}}{,}{a}{=}{b}{+}\frac{{1}}{{2}}{\wedge }\mathrm{c__1}{=}\frac{{1}}{{2}}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left(\frac{\mathrm{z__2}{+}{1}{-}\mathrm{z__1}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}}{{2}{}\mathrm{z__2}}\right)}^{{a}}{}{\left(\frac{\mathrm{z__1}{+}{1}{-}\mathrm{z__2}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}}{{2}{}\mathrm{z__1}}\right)}^{{a}}{}{}_{{2}}{F}_{{1}}{}\left({a}{,}{a}{-}{b}{+}{1}{;}{b}{;}\frac{{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}^{{2}}}{{4}{}\mathrm{z__1}{}\mathrm{z__2}}\right){,}\mathrm{c__1}{=}{b}{\wedge }\mathrm{c__2}{=}{b}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\left(\frac{\mathrm{z__2}{+}{1}{-}\mathrm{z__1}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}}{{2}{}\mathrm{z__2}}\right)}^{{b}}{}{\left(\frac{\mathrm{z__1}{+}{1}{-}\mathrm{z__2}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}}{{2}{}\mathrm{z__1}}\right)}^{{b}}{}{}_{{2}}{F}_{{1}}{}\left({b}{,}{b}{-}{a}{+}{1}{;}{a}{;}\frac{{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}^{{2}}}{{4}{}\mathrm{z__1}{}\mathrm{z__2}}\right){,}\mathrm{c__1}{=}{a}{\wedge }\mathrm{c__2}{=}{a}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}{b}{;}{a}{-}{b}{+}{1}{;}\frac{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right){}\left(\mathrm{z__1}{+}{1}{-}\mathrm{z__2}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}{{2}{}\mathrm{z__1}{}\left({-}\mathrm{z__2}{-}{1}{+}\mathrm{z__1}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}\right){}{\left(\frac{\mathrm{z__1}{+}{1}{-}\mathrm{z__2}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}}{{2}{}\mathrm{z__1}}\right)}^{{a}}{,}\mathrm{c__1}{=}{a}{-}{b}{+}{1}{\wedge }\mathrm{c__2}{=}{b}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}{b}{;}{a}{-}{b}{+}{1}{;}\frac{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right){}\left({-}\mathrm{z__2}{-}{1}{+}\mathrm{z__1}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}{{2}{}\mathrm{z__2}{}\left(\mathrm{z__1}{+}{1}{-}\mathrm{z__2}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}\right){}{\left(\frac{\mathrm{z__2}{+}{1}{-}\mathrm{z__1}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}}{{2}{}\mathrm{z__2}}\right)}^{{a}}{,}\mathrm{c__2}{=}{a}{-}{b}{+}{1}{\wedge }\mathrm{c__1}{=}{b}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}{b}{;}{b}{-}{a}{+}{1}{;}\frac{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right){}\left(\mathrm{z__1}{+}{1}{-}\mathrm{z__2}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}{{2}{}\mathrm{z__1}{}\left({-}\mathrm{z__2}{-}{1}{+}\mathrm{z__1}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}\right){}{\left(\frac{\mathrm{z__1}{+}{1}{-}\mathrm{z__2}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}}{{2}{}\mathrm{z__1}}\right)}^{{b}}{,}\mathrm{c__1}{=}{b}{-}{a}{+}{1}{\wedge }\mathrm{c__2}{=}{a}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}{b}{;}{b}{-}{a}{+}{1}{;}\frac{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right){}\left({-}\mathrm{z__2}{-}{1}{+}\mathrm{z__1}{+}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}{{2}{}\mathrm{z__2}{}\left(\mathrm{z__1}{+}{1}{-}\mathrm{z__2}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}\right)}\right){}{\left(\frac{\mathrm{z__2}{+}{1}{-}\mathrm{z__1}{-}\sqrt{{-}{4}{}\mathrm{z__1}{}\mathrm{z__2}{+}{\left({-}{1}{+}\mathrm{z__1}{+}\mathrm{z__2}\right)}^{{2}}}}{{2}{}\mathrm{z__2}}\right)}^{{b}}{,}\mathrm{c__2}{=}{b}{-}{a}{+}{1}{\wedge }\mathrm{c__1}{=}{a}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left({a}{,}{b}{,}\frac{\mathrm{c__1}}{{2}}{+}\frac{\mathrm{c__2}}{{2}}{,}\frac{\mathrm{c__1}}{{2}}{+}\frac{\mathrm{c__2}}{{2}}{-}\frac{{1}}{{2}}{;}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{c__1}{+}\mathrm{c__2}{-}{1}{;}{4}{}\mathrm{z__1}\right){,}\mathrm{z__1}{=}\mathrm{z__2}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{4}}{F}_{{3}}{}\left(\frac{{a}}{{2}}{,}\frac{{b}}{{2}}{,}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{,}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}{;}\mathrm{c__1}{,}\frac{\mathrm{c__1}}{{2}}{,}\frac{\mathrm{c__1}}{{2}}{+}\frac{{1}}{{2}}{;}{-}{4}{}{\mathrm{z__1}}^{{2}}\right){,}\mathrm{z__2}{=}{-}\mathrm{z__1}{\wedge }\mathrm{c__1}{=}\mathrm{c__2}\right]$ (8)

Other special values of AppellF4 can be seen using FunctionAdvisor(special_values, AppellF4).

By requesting the sum form of AppellF4, 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:

 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{AppellF4}\right)$
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{m}{+}{n}}{}{\left({b}\right)}_{{m}{+}{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left(\mathrm{c__1}\right)}_{{m}}{}{\left(\mathrm{c__2}\right)}_{{n}}{}{m}{!}{}{n}{!}}{,}\sqrt{\left|\mathrm{z__2}\right|}{+}\sqrt{\left|\mathrm{z__1}\right|}{<}{1}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left({b}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}{b}{+}{k}{;}\mathrm{c__2}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left(\mathrm{c__1}\right)}_{{k}}{}{k}{!}}{,}\sqrt{\left|\mathrm{z__2}\right|}{+}\sqrt{\left|\mathrm{z__1}\right|}{<}{1}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left({b}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}{b}{+}{k}{;}\mathrm{c__1}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left(\mathrm{c__2}\right)}_{{k}}{}{k}{!}}{,}\sqrt{\left|\mathrm{z__2}\right|}{+}\sqrt{\left|\mathrm{z__1}\right|}{<}{1}\right]$ (9)

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

AppellF4 admits identities analogous to Euler identities for the hypergeometric function. These Euler-type identities, as well as contiguity identities, are visible using the FunctionAdvisor with the option identities, or directly from the function. For example,

 >
 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({b}{-}{a}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{a}}{}{{F}}_{{4}}{}\left({a}{,}{a}{-}\mathrm{c__2}{+}{1}{,}{a}{-}{b}{+}{1}{,}\mathrm{c__1}{,}\frac{{1}}{\mathrm{z__2}}{,}\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{a}\right){}{\mathrm{\Gamma }}{}\left({b}\right)}{+}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({a}{-}{b}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{b}}{}{{F}}_{{4}}{}\left({b}{,}{b}{-}\mathrm{c__2}{+}{1}{,}{b}{-}{a}{+}{1}{,}\mathrm{c__1}{,}\frac{{1}}{\mathrm{z__2}}{,}\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{b}\right){}{\mathrm{\Gamma }}{}\left({a}\right)}$ (10)

Among other situations, this identity is useful when the sum of the square roots of the absolute values of ${z}_{1}$ and ${z}_{2}$ is larger than 1 but the same sum constructed with the arguments in the same position of AppellF4 on the right-hand side is smaller than 1. Another case where this identity is useful is when ${z}_{1}=1$, so that the two AppellF4 functions on the right-hand side will have the two main variables (last arguments) equal, in turn a special value of hypergeometric 4F3 type:

 >
 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}{1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({b}{-}{a}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{a}}{}{}_{{4}}{F}_{{3}}{}\left({a}{,}{a}{-}\mathrm{c__2}{+}{1}{,}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}{+}\frac{\mathrm{c__1}}{{2}}{,}\frac{{a}}{{2}}{-}\frac{{b}}{{2}}{+}\frac{{1}}{{2}}{+}\frac{\mathrm{c__1}}{{2}}{;}\mathrm{c__1}{,}{a}{-}{b}{+}{1}{,}{a}{-}{b}{+}\mathrm{c__1}{;}\frac{{4}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{a}\right){}{\mathrm{\Gamma }}{}\left({b}\right)}{+}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({a}{-}{b}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{b}}{}{}_{{4}}{F}_{{3}}{}\left({b}{,}{b}{-}\mathrm{c__2}{+}{1}{,}\frac{{b}}{{2}}{-}\frac{{a}}{{2}}{+}\frac{\mathrm{c__1}}{{2}}{,}\frac{{b}}{{2}}{-}\frac{{a}}{{2}}{+}\frac{{1}}{{2}}{+}\frac{\mathrm{c__1}}{{2}}{;}\mathrm{c__1}{,}{b}{-}{a}{+}{1}{,}{b}{-}{a}{+}\mathrm{c__1}{;}\frac{{4}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{b}\right){}{\mathrm{\Gamma }}{}\left({a}\right)}$ (11)

This formula analytically extends to the whole complex plane the AppellF4 series when any of ${z}_{1}=1$ or ${z}_{2}=1$ (the latter using the symmetry of AppellF4 - see the beginning of the Description section).

A contiguity transformation for AppellF4

 >
 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{4}}{}\left({b}{,}{a}{+}{1}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}{a}}{{a}{-}{b}}{-}\frac{{b}{}{{F}}_{{4}}{}\left({a}{,}{b}{+}{1}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{a}{-}{b}}$ (12)

The contiguity transformations available in this way are

 > $\mathrm{indices}\left(\mathrm{AppellF4}:-\mathrm{Transformations}\left["Contiguity"\right]\right)$
 $\left[{1}\right]{,}\left[{2}\right]{,}\left[{3}\right]{,}\left[{4}\right]$ (13)

By using differential algebra techniques, the PDE system satisfied by AppellF4 can be transformed into an equivalent PDE system where one of the equations is a fourth order linear ODE in ${z}_{2}$ parametrized by ${z}_{1}$. This linear ODE has four regular singularities, some of which depend on ${z}_{1}$ and the function's parameters. These singularities can be see directly from the function using the MathematicalFunctions:-Evalf:-Singularities command

 > $\mathrm{MathematicalFunctions}:-\mathrm{Evalf}:-\mathrm{Singularities}\left(\mathrm{AppellF4}\left(a,b,\mathrm{c__1},\mathrm{c__2},\mathrm{z__1},\mathrm{z__2}\right)\right)$
 $\left[{0}{,}\frac{\left(\mathrm{z__1}{-}{1}\right){}\left({a}{+}{b}{-}\mathrm{c__1}{+}{1}\right){}\left({a}{+}{b}{-}\mathrm{c__1}{-}{2}{}\mathrm{c__2}{+}{3}\right)}{\left(\mathrm{c__1}{-}{1}{-}{b}{+}{a}\right){}\left({-}\mathrm{c__1}{+}{1}{-}{b}{+}{a}\right)}{,}\mathrm{z__1}{+}{1}{-}{2}{}\sqrt{\mathrm{z__1}}{,}\mathrm{z__1}{+}{1}{+}{2}{}\sqrt{\mathrm{z__1}}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]$ (14)

You can also see a general presentation of AppellF4, organized into sections and including plots, using the FunctionAdvisor

 > $\mathrm{FunctionAdvisor}\left(\mathrm{AppellF4}\right)$

AppellF4

describe

 ${\mathrm{AppellF4}}{=}{\mathrm{Appell 2-variable hypergeometric function F4}}$

definition

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left({b}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left(\mathrm{c__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{c__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}$ $\sqrt{\left|\mathrm{z__1}\right|}{+}\sqrt{\left|\mathrm{z__2}\right|}{<}{1}$

classify function

 ${\mathrm{Appell}}$

symmetries

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__2}{,}\mathrm{c__1}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ ${{F}}_{{4}}{}\left({b}{,}{a}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$

plot

singularities

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left(\mathrm{c__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{b}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right)\right){\vee }\left(\mathrm{c__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{b}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{a}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{b}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{b}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__1}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{b}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{<}\mathrm{c__1}{\wedge }{b}{<}\mathrm{c__1}\right){\vee }\left(\mathrm{c__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{b}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right)\right){\vee }\left(\mathrm{c__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{b}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{a}{<}\mathrm{c__2}\right){\vee }\left(\mathrm{c__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{b}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{b}{<}\mathrm{c__2}\right){\vee }\left(\mathrm{c__2}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{b}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right){\wedge }{a}{<}\mathrm{c__2}{\wedge }{b}{<}\mathrm{c__2}\right)$

branch points

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left({a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{b}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{z__1}{\in }\left[{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]\right){\vee }\left({a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{b}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{z__2}{\in }\left[{1}{,}{\mathrm{\infty }}{+}{\mathrm{\infty }}{}{I}\right]\right)$

branch cuts

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\left({a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{b}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{1}{<}\mathrm{z__1}\right){\vee }\left({a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{b}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{1}{<}\mathrm{z__2}\right)$

special values

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ $\mathrm{z__1}{=}{0}{\wedge }\mathrm{z__2}{=}{0}$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ ${a}{=}{0}$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{1}$ ${b}{=}{0}$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}{b}{;}\mathrm{c__2}{;}\mathrm{z__2}\right)$ $\mathrm{z__1}{=}{0}$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{}_{{2}}{F}_{{1}}{}\left({a}{,}{b}{;}\mathrm{c__1}{;}\mathrm{z__1}\right)$ $\mathrm{z__2}{=}{0}$

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

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

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

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

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

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

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

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

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

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

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

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

identities

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

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{4}}{}\left({b}{,}{a}{+}{n}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}\frac{{b}{}\mathrm{z__1}{}\left({\sum }_{{k}{=}{1}}^{{n}}{}{{F}}_{{4}}{}\left({a}{+}{k}{,}{b}{+}{1}{,}\mathrm{c__1}{+}{1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{c__1}}{-}\frac{{b}{}\mathrm{z__2}{}\left({\sum }_{{k}{=}{1}}^{{n}}{}{{F}}_{{4}}{}\left({a}{+}{k}{,}{b}{+}{1}{,}\mathrm{c__1}{,}\mathrm{c__2}{+}{1}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right)}{\mathrm{c__2}}$ $\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__2}{\ne }{1}{\wedge }\mathrm{c__1}{\ne }{0}{\wedge }\mathrm{c__2}{\ne }{0}$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{4}}{}\left({b}{,}{a}{+}{n}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}\left({\sum }_{{i}{=}{0}}^{{n}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{k}{=}{0}}^{{n}{-}{i}}{}\frac{{{F}}_{{4}}{}\left({a}{+}{i}{+}{k}{,}{k}{+}{i}{+}{b}{,}{i}{+}\mathrm{c__1}{,}{k}{+}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){}\left(\genfrac{}{}{0}{}{{n}}{{i}}\right){}\left(\genfrac{}{}{0}{}{{n}{-}{i}}{{k}}\right){}{\left({b}\right)}_{{k}{+}{i}}{}{\mathrm{z__1}}^{{i}}{}{\mathrm{z__2}}^{{k}}}{{\left(\mathrm{c__1}\right)}_{{i}}{}{\left(\mathrm{c__2}\right)}_{{k}}}\right){+}{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)$ $\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__2}{\ne }{1}{\wedge }\left(\mathrm{c__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\vee }{n}{\le }\left|\mathrm{c__1}\right|\right){\wedge }\left(\mathrm{c__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\vee }{n}{\le }\left|\mathrm{c__2}\right|\right)$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{-}{n}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){-}{a}{}{b}{}\mathrm{z__1}{}\left({\sum }_{{k}{=}{0}}^{{n}{-}{1}}{}\frac{{{F}}_{{4}}{}\left({a}{+}{1}{,}{b}{+}{1}{,}\mathrm{c__1}{+}{1}{-}{k}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{\left(\mathrm{c__1}{-}{k}\right){}\left(\mathrm{c__1}{-}{k}{-}{1}\right)}\right)$ $\mathrm{z__1}{\ne }{1}{\wedge }\mathrm{z__2}{\ne }{1}{\wedge }\left(\mathrm{c__1}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonnegint}}{'}\right]\right)\right){\vee }{n}{-}{1}{<}\mathrm{c__1}\right)$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({b}{-}{a}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{a}}{}{{F}}_{{4}}{}\left({a}{,}{a}{-}\mathrm{c__2}{+}{1}{,}{a}{-}{b}{+}{1}{,}\mathrm{c__1}{,}\frac{{1}}{\mathrm{z__2}}{,}\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{a}\right){}{\mathrm{\Gamma }}{}\left({b}\right)}{+}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({-}{b}{+}{a}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{b}}{}{{F}}_{{4}}{}\left({b}{,}{b}{-}\mathrm{c__2}{+}{1}{,}{b}{-}{a}{+}{1}{,}\mathrm{c__1}{,}\frac{{1}}{\mathrm{z__2}}{,}\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{b}\right){}{\mathrm{\Gamma }}{}\left({a}\right)}$ $\mathrm{z__2}{\ne }{0}{\wedge }{a}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }{b}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\mathrm{c__2}{::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\left(\mathrm{c__2}{-}{a}\right){::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\left(\mathrm{c__2}{-}{b}\right){::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)\right){\wedge }\left({b}{-}{a}\right){::}\left({¬}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{integer}}{'}\right]\right)\right)$

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

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

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

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

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{2}}{}\left({a}{,}{b}{,}\frac{{1}}{{2}}{+}{a}{-}{b}{,}\mathrm{c__1}{,}{2}{}{a}{+}{1}{-}{2}{}{b}{,}\frac{\mathrm{z__1}}{{\left(\sqrt{\mathrm{z__2}}{+}{1}\right)}^{{2}}}{,}\frac{{4}{}\sqrt{\mathrm{z__2}}}{{\left(\sqrt{\mathrm{z__2}}{+}{1}\right)}^{{2}}}\right)}{{\left(\sqrt{\mathrm{z__2}}{+}{1}\right)}^{{2}{}{a}}}$ $\mathrm{c__2}{=}{a}{-}{b}{+}{1}$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{2}}{}\left({b}{,}{a}{,}\frac{{1}}{{2}}{+}{b}{-}{a}{,}\mathrm{c__1}{,}{2}{}{b}{+}{1}{-}{2}{}{a}{,}\frac{\mathrm{z__1}}{{\left(\sqrt{\mathrm{z__2}}{+}{1}\right)}^{{2}}}{,}\frac{{4}{}\sqrt{\mathrm{z__2}}}{{\left(\sqrt{\mathrm{z__2}}{+}{1}\right)}^{{2}}}\right)}{{\left(\sqrt{\mathrm{z__2}}{+}{1}\right)}^{{2}{}{b}}}$ $\mathrm{c__2}{=}{b}{-}{a}{+}{1}$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{2}}{}\left({a}{,}{b}{,}\frac{{1}}{{2}}{+}{a}{-}{b}{,}\mathrm{c__2}{,}{2}{}{a}{+}{1}{-}{2}{}{b}{,}\frac{\mathrm{z__2}}{{\left(\sqrt{\mathrm{z__1}}{+}{1}\right)}^{{2}}}{,}\frac{{4}{}\sqrt{\mathrm{z__1}}}{{\left(\sqrt{\mathrm{z__1}}{+}{1}\right)}^{{2}}}\right)}{{\left(\sqrt{\mathrm{z__1}}{+}{1}\right)}^{{2}{}{a}}}$ $\mathrm{c__1}{=}{a}{-}{b}{+}{1}$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{{F}}_{{2}}{}\left({b}{,}{a}{,}\frac{{1}}{{2}}{+}{b}{-}{a}{,}\mathrm{c__2}{,}{2}{}{b}{+}{1}{-}{2}{}{a}{,}\frac{\mathrm{z__2}}{{\left(\sqrt{\mathrm{z__1}}{+}{1}\right)}^{{2}}}{,}\frac{{4}{}\sqrt{\mathrm{z__1}}}{{\left(\sqrt{\mathrm{z__1}}{+}{1}\right)}^{{2}}}\right)}{{\left(\sqrt{\mathrm{z__1}}{+}{1}\right)}^{{2}{}{b}}}$ $\mathrm{c__1}{=}{b}{-}{a}{+}{1}$

 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{Physics}}{:-}{\mathrm{Library}}{:-}{\mathrm{Add}}{}\left(\frac{{\left({a}\right)}_{{\mathrm{k1}}{+}{\mathrm{k2}}}{}{{F}}_{{2}}{}\left({b}{,}{-}{\mathrm{k1}}{,}{-}{\mathrm{k2}}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)}{{\mathrm{k1}}{!}{}{\mathrm{k2}}{!}}{,}{\mathrm{k1}}{+}{\mathrm{k2}}{\le }{-}{a}\right)}{{\left({-1}\right)}^{{a}}}$ ${a}{::}{\mathrm{Typesetting:-_Hold}}{}\left(\left[{'}{\mathrm{nonposint}}{'}\right]\right)$

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

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

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