Appell Functions - Maple Help

The four Appell functions

The four multi-parameter Appell functions, AppellF1, AppellF2, AppellF3, and AppellF4

Description

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

Definition and Symmetries

 The definition of the four Appell series and the corresponding domains of convergence can be seen through the FunctionAdvisor. For example, for AppellF1,
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left({c}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]$ (1)
 From this definition, by swapping the AppellF1 variables subscripted with the numbers 1 and 2, the function remains the same; hence
 > $\mathrm{FunctionAdvisor}\left(\mathrm{symmetries},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right]$ (2)
 Analogously, the definition and symmetry of AppellF2 are
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{AppellF2}\right)$
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left(\mathrm{c__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{c__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\left|\mathrm{z__1}\right|{+}\left|\mathrm{z__2}\right|{<}{1}\right]$ (3)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{symmetries},\mathrm{AppellF2}\right)$
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}\mathrm{c__2}{,}\mathrm{c__1}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right]$ (4)
 The cases of AppellF3 and AppellF4 are more general in that, from their definition, besides the symmetry under a swap of the subscripted variables, these two functions have additional symmetries under exchange of positions of the function's parameters
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{a__2}\right)}_{{\mathrm{_k2}}}{}{\left(\mathrm{b__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{b__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left({c}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]$ (5)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{symmetries},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__2}{,}\mathrm{a__1}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}{c}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}{{F}}_{{3}}{}\left(\mathrm{b__1}{,}\mathrm{a__2}{,}\mathrm{a__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{b__2}{,}\mathrm{b__1}{,}\mathrm{a__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right]$ (6)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{definition},\mathrm{AppellF4}\right)$
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{\mathrm{_k1}}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{\mathrm{_k2}}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\left({b}\right)}_{{\mathrm{_k1}}{+}{\mathrm{_k2}}}{}{\mathrm{z__1}}^{{\mathrm{_k1}}}{}{\mathrm{z__2}}^{{\mathrm{_k2}}}}{{\left(\mathrm{c__1}\right)}_{{\mathrm{_k1}}}{}{\left(\mathrm{c__2}\right)}_{{\mathrm{_k2}}}{}{\mathrm{_k1}}{!}{}{\mathrm{_k2}}{!}}{,}\sqrt{\left|\mathrm{z__1}\right|}{+}\sqrt{\left|\mathrm{z__2}\right|}{<}{1}\right]$ (7)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{symmetries},\mathrm{AppellF4}\right)$
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__2}{,}\mathrm{c__1}{,}\mathrm{z__2}{,}\mathrm{z__1}\right){=}{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}{{F}}_{{4}}{}\left({b}{,}{a}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right)\right]$ (8)

Polynomial and Singular Cases

 From these four definitions, the 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: these pochhammer functions will be equal to zero when the summation indices of these series are bigger than the absolute values of the $c$ parameters.
 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. As is the case of the hypergeometric function, when both the pochhammers in the numerators and denominators have non-positive integer arguments, the Appell functions are polynomial when the absolute values of the non-positive integers in the numerators are smaller than or equal to the absolute values of the non-positive integers in the denominators, and singular otherwise. The combinatorial of all these conditions can also be consulted using the FunctionAdvisor. For example, for AppellF1, the singular cases happen when any of the following conditions hold
 > $\mathrm{FunctionAdvisor}\left(\mathrm{singularities},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){,}\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{a}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }{a}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{<}{c}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__2}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right){\vee }\left({c}{::}{'}{\mathrm{nonposint}}{'}{\wedge }{a}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right){\wedge }\mathrm{b__1}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__2}{::}{'}{\mathrm{nonposint}}{'}{\wedge }\mathrm{b__1}{+}\mathrm{b__2}{<}{c}\right)\right]$ (9)
 The conditions for both the singular and the polynomial cases can also be seen from the any of the four Appell functions. For example, the same conditions for the singular cases of AppellF1 can be seen entering AppellF1:-Singularities(), so with no arguments, and in the same way the conditions for the six polynomial cases of AppellF1 are
 > $\mathrm{AppellF1}:-\mathrm{SpecialValues}:-\mathrm{Polynomial}\left(\right)$
 ${6}{,}\left({a}{,}{\mathrm{b1}}{,}{\mathrm{b2}}{,}{c}{,}{\mathrm{z1}}{,}{\mathrm{z2}}\right){↦}'\left[\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{a}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{\le }{a}\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{\le }{\mathrm{b1}}{+}{\mathrm{b2}}\right]{,}\left[{\mathrm{b1}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]{,}\left[{\mathrm{b2}}{::}{'}{\mathrm{nonposint}}{'}{,}{c}{::}\left({¬}{'}{\mathrm{nonposint}}{'}\right)\right]\right]'$ (10)

Single Power Series with Hypergeometric Coefficients

 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 AppellF1 and AppellF3,
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{AppellF1}\right)$
 $\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{m}{+}{n}}{}{\left(\mathrm{b__1}\right)}_{{m}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left({c}\right)}_{{m}{+}{n}}{}{m}{!}{}{n}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__1}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__2}{;}{c}{+}{k}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}\right]{,}\left[{{F}}_{{1}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__2}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__1}{;}{c}{+}{k}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__2}\right|{<}{1}\right]$ (11)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{m}}{}{\left(\mathrm{a__2}\right)}_{{n}}{}{\left(\mathrm{b__1}\right)}_{{m}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left({c}\right)}_{{m}{+}{n}}{}{m}{!}{}{n}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}{\wedge }\left|\mathrm{z__2}\right|{<}{1}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__1}\right)}_{{k}}{}{\left(\mathrm{b__1}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{+}{k}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__1}\right|{<}{1}\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left(\mathrm{a__2}\right)}_{{k}}{}{\left(\mathrm{b__2}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{+}{k}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left({c}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__2}\right|{<}{1}\right]$ (12)
 As indicated in the formulas above, for these two Appell functions the domain of convergence of the single sum with hypergeometric coefficients is larger than the domain of convergence of the double series, because the hypergeometric coefficient in the single sum - say the one in ${z}_{2}$ - analytically extends the series with regards to the other variable - say ${z}_{1}$ - entering the hypergeometric coefficient. Hence, for AppellF1 and AppellF3, the case where one of the two variables, ${z}_{1}$ or ${z}_{2}$, is equal to 1, is convergent only when the corresponding hypergeometric coefficient in the single sum form is convergent. For instance, for AppellF1 the convergent case at ${z}_{1}=1$ requires that $0<-\mathrm{Re}\left(-c+a+{b}_{1}\right)$.
 The situation is different for AppellF2 and AppellF4, where 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:
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{AppellF2}\right)$
 $\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{m}{+}{n}}{}{\left(\mathrm{b__1}\right)}_{{m}}{}{\left(\mathrm{b__2}\right)}_{{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left(\mathrm{c__1}\right)}_{{m}}{}{\left(\mathrm{c__2}\right)}_{{n}}{}{m}{!}{}{n}{!}}{,}\left|\mathrm{z__2}\right|{+}\left|\mathrm{z__1}\right|{<}{1}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__1}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__2}{;}\mathrm{c__2}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left(\mathrm{c__1}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__2}\right|{+}\left|\mathrm{z__1}\right|{<}{1}\right]{,}\left[{{F}}_{{2}}{}\left({a}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left(\mathrm{b__2}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}\mathrm{b__1}{;}\mathrm{c__1}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left(\mathrm{c__2}\right)}_{{k}}{}{k}{!}}{,}\left|\mathrm{z__2}\right|{+}\left|\mathrm{z__1}\right|{<}{1}\right]$ (13)
 > $\mathrm{FunctionAdvisor}\left(\mathrm{sum_form},\mathrm{AppellF4}\right)$
 $\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{m}{=}{0}}^{{\mathrm{\infty }}}{}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{m}{+}{n}}{}{\left({b}\right)}_{{m}{+}{n}}{}{\mathrm{z__1}}^{{m}}{}{\mathrm{z__2}}^{{n}}}{{\left(\mathrm{c__1}\right)}_{{m}}{}{\left(\mathrm{c__2}\right)}_{{n}}{}{m}{!}{}{n}{!}}{,}\sqrt{\left|\mathrm{z__2}\right|}{+}\sqrt{\left|\mathrm{z__1}\right|}{<}{1}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left({b}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}{b}{+}{k}{;}\mathrm{c__2}{;}\mathrm{z__2}\right){}{\mathrm{z__1}}^{{k}}}{{\left(\mathrm{c__1}\right)}_{{k}}{}{k}{!}}{,}\sqrt{\left|\mathrm{z__2}\right|}{+}\sqrt{\left|\mathrm{z__1}\right|}{<}{1}\right]{,}\left[{{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}{\sum }_{{k}{=}{0}}^{{\mathrm{\infty }}}{}\frac{{\left({a}\right)}_{{k}}{}{\left({b}\right)}_{{k}}{}{}_{{2}}{F}_{{1}}{}\left({a}{+}{k}{,}{b}{+}{k}{;}\mathrm{c__1}{;}\mathrm{z__1}\right){}{\mathrm{z__2}}^{{k}}}{{\left(\mathrm{c__2}\right)}_{{k}}{}{k}{!}}{,}\sqrt{\left|\mathrm{z__2}\right|}{+}\sqrt{\left|\mathrm{z__1}\right|}{<}{1}\right]$ (14)
 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.

Analytic Extension from the Appell Series to the Appell Functions

 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 AppellF3
 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral_form},\mathrm{AppellF3}\right)$
 $\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{\left({1}{-}{u}\right)}^{{-}{1}{+}\mathrm{b__1}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__2}{,}\mathrm{b__2}{;}{c}{-}\mathrm{b__1}{;}\mathrm{z__2}{}{u}\right)}{{{u}}^{{-}{c}{+}\mathrm{b__1}{+}{1}}{}{\left({1}{+}\left({u}{-}{1}\right){}\mathrm{z__1}\right)}^{\mathrm{a__1}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{u}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{b__1}\right){}{\mathrm{\Gamma }}{}\left({c}{-}\mathrm{b__1}\right)}{,}{0}{<}{\mathrm{\Re }}{}\left(\mathrm{b__1}\right){\wedge }{0}{<}{\mathrm{\Re }}{}\left({c}\right){\wedge }{0}{<}{-}{\mathrm{\Re }}{}\left({-}{c}{+}\mathrm{b__1}\right)\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}\frac{{\left({1}{-}{u}\right)}^{\mathrm{b__2}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{c}{-}\mathrm{b__2}{;}{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)\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}{{u}}^{{\mathrm{\rho }}{-}{1}}{}{\left({1}{-}{u}\right)}^{{c}{-}{\mathrm{\rho }}{-}{1}}{}{}_{{2}}{F}_{{1}}{}\left(\mathrm{a__1}{,}\mathrm{b__1}{;}{\mathrm{\rho }}{;}{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)\right]{,}\left[{{F}}_{{3}}{}\left(\mathrm{a__1}{,}\mathrm{a__2}{,}\mathrm{b__1}{,}\mathrm{b__2}{,}{c}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left({c}\right){}\left({{\int }}_{{0}}^{{1}}{{\int }}_{{0}}^{{1}{-}{v}}\frac{{{u}}^{{-}{1}{+}\mathrm{b__1}}{}{{v}}^{\mathrm{b__2}{-}{1}}}{{\left({1}{-}{u}{-}{v}\right)}^{{-}{c}{+}\mathrm{b__1}{+}\mathrm{b__2}{+}{1}}{}{\left({-}{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)\right]$ (15)
 In the case of AppellF4, single integral representation exists only for particular values of the function's parameters, for example two cases are
 > $\mathrm{FunctionAdvisor}\left(\mathrm{integral_form},\mathrm{AppellF4}\right)\left[1..2\right]$
 $\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]$ (16)
 All these integral representations are the starting point for the derivation of many of the identities known for the four Appell functions.

Euler-Type and Contiguity Identities

 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,
 >
 ${{F}}_{{4}}{}\left({a}{,}{b}{,}\mathrm{c__1}{,}\mathrm{c__2}{,}\mathrm{z__1}{,}\mathrm{z__2}\right){=}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({b}{-}{a}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{a}}{}{{F}}_{{4}}{}\left({a}{,}{a}{-}\mathrm{c__2}{+}{1}{,}{a}{-}{b}{+}{1}{,}\mathrm{c__1}{,}\frac{{1}}{\mathrm{z__2}}{,}\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{a}\right){}{\mathrm{\Gamma }}{}\left({b}\right)}{+}\frac{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}\right){}{\mathrm{\Gamma }}{}\left({a}{-}{b}\right){}{\left({-}\mathrm{z__2}\right)}^{{-}{b}}{}{{F}}_{{4}}{}\left({b}{,}{1}{+}{b}{-}\mathrm{c__2}{,}{b}{-}{a}{+}{1}{,}\mathrm{c__1}{,}\frac{{1}}{\mathrm{z__2}}{,}\frac{\mathrm{z__1}}{\mathrm{z__2}}\right)}{{\mathrm{\Gamma }}{}\left(\mathrm{c__2}{-}{b}\right){}{\mathrm{\Gamma }}{}\left({a}\right)}$ (17)
 and this identity can be used to evaluate Appell