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AppellF3

The AppellF3 function

	Calling Sequence
	AppellF3( $a_{1}, a_{2}, b_{1}, b_{2}, c, z_{1}, z_{2}$ )

Parameters

$a_{1}$	-	algebraic expression
$a_{2}$	-	algebraic expression
$b_{1}$	-	algebraic expression
$b_{2}$	-	algebraic expression
$c$	-	algebraic expression
$z_{1}$	-	algebraic expression
$z_{2}$	-	algebraic expression

Description

•

As is the case of all the four multi-parameter Appell functions, AppellF3, is a doubly hypergeometric function that includes as particular cases the 2F1 hypergeometric and some cases of the MeijerG function, and with them most of the known functions of mathematical physics. Among other situations, AppellF3 appears in the solution to differential equations in general relativity, quantum mechanics, and molecular and atomic physics.

Initialization: Set the display of special functions in output to typeset mathematical notation (textbook notation):

>	$Typesetting :- EnableTypesetRule (Typesetting :- SpecialFunctionRules) &colon;$

The definition of the AppellF3 series and the corresponding domain of convergence can be seen through the FunctionAdvisor

>	$FunctionAdvisor (definition, AppellF3)$

$[F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \sum_{_k1 = 0}^{\infty} \sum_{_k2 = 0}^{\infty} \frac{{(a__1)}_{_k1} {(a__2)}_{_k2} {(b__1)}_{_k1} {(b__2)}_{_k2} {z__1}^{_k1} {z__2}^{_k2}}{{(c)}_{_k1 +_k2}_k1!_k2!}, |z__1| < 1 \land |z__2| < 1]$

(1)

A distinction is made between the AppellF3 doubly hypergeometric series, with the restricted domain of convergence shown above, and the AppellF3 function, that coincides with the series in its domain of convergence but also extends it analytically to the whole complex plane.

From the definition above, by swapping the AppellF3 variables subscripted with the numbers 1 and 2, the function remains the same; hence

>	$FunctionAdvisor (symmetries, AppellF3)$

$[F_{3} (a__2, a__1, b__2, b__1, c, z__2, z__1) = F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2), F_{3} (b__1, a__2, a__1, b__2, c, z__1, z__2) = F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2), F_{3} (a__1, b__2, b__1, a__2, c, z__1, z__2) = F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2)]$

(2)

Note the existence of other symmetries, also visible in the double sum definition.

From the series' definition, AppellF3 is singular (division by zero) when the $c$ parameter entering the pochhammer function in the denominator of the series is a non-positive integer because the pochhammer function will be equal to zero when the summation index of the series is bigger than the absolute value of $c$ .

For an analogous reason, when the $a_{1}$ and/or $a_{2}$ and/or $b_{1}$ and/or $b_{2}$ parameters entering the pochhammer functions in the numerator of the series are non-positive integers, the series will truncate and AppellF3 will be polynomial in one of the two of $z_{1}, z_{2}$ . As is the case of the hypergeometric function, when the pochhammers in both the numerator and the denominator have non-positive integer arguments, AppellF3 is polynomial if the absolute value of the non-positive integers in the pochhammers of the numerator are smaller than or equal to the absolute value of the non-positive integer (parameter $c$ ) in the pochhammer in the denominator, and singular otherwise. Consult the FunctionAdvisor for comprehensive information on the combinations of all these conditions. For example, the singular cases happen when any of the following conditions hold

>	$FunctionAdvisor (singularities, AppellF3)$

$[F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2), (c :: ℤ^{(0, -)} \land a__1 :: (\neg ℤ^{(0, -)}) \land b__1 :: (\neg ℤ^{(0, -)})) \lor (c :: ℤ^{(0, -)} \land a__2 :: (\neg ℤ^{(0, -)}) \land b__2 :: (\neg ℤ^{(0, -)})) \lor (c :: ℤ^{(0, -)} \land a__1 :: (\neg ℤ^{(0, -)}) \land a__2 :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land b__2 :: (\neg ℤ^{(0, -)}) \land b__1 + a__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: (\neg ℤ^{(0, -)}) \land a__2 :: (\neg ℤ^{(0, -)}) \land b__1 :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land b__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: (\neg ℤ^{(0, -)}) \land a__2 :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land b__1 + a__2 < c \land b__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: ℤ^{(0, -)} \land b__1 :: (\neg ℤ^{(0, -)}) \land b__2 :: (\neg ℤ^{(0, -)}) \land a__1 + a__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: (\neg ℤ^{(0, -)}) \land b__1 :: (\neg ℤ^{(0, -)}) \land b__2 :: ℤ^{(0, -)} \land a__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: ℤ^{(0, -)} \land b__1 :: (\neg ℤ^{(0, -)}) \land b__2 :: ℤ^{(0, -)} \land a__1 + a__2 < c \land a__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land a__1 + a__2 < c \land a__1 + b__2 < c \land b__1 + a__2 < c \land b__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: (\neg ℤ^{(0, -)}) \land b__1 :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land a__1 + b__2 < c \land b__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land b__2 :: (\neg ℤ^{(0, -)}) \land a__1 + a__2 < c \land b__1 + a__2 < c)]$

(3)

The AppellF3 series is analytically extended to the AppellF3 function defined over the whole complex plane using identities and mainly by integral representations in terms of Eulerian integrals:

>	$FunctionAdvisor (integral_form, AppellF3)$

$[F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \frac{Γ (c) (\int_{0}^{1} \frac{{(1 - u)}^{- 1 + b__1} {}_{2}F_{1} (a__2, b__2; c - b__1; z__2 u)}{u^{- c + b__1 + 1} {(1 + (u - 1) z__1)}^{a__1}} &DifferentialD; u)}{Γ (b__1) Γ (c - b__1)}, 0 < ℜ (b__1) \land 0 < ℜ (c) \land 0 < - ℜ (- c + b__1)], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \frac{Γ (c) (\int_{0}^{1} \frac{{(1 - u)}^{b__2 - 1} {}_{2}F_{1} (a__1, b__1; c - b__2; z__1 u)}{u^{- c + b__2 + 1} {(1 + (u - 1) z__2)}^{a__2}} &DifferentialD; u)}{Γ (b__2) Γ (c - b__2)}, 0 < ℜ (b__2) \land 0 < ℜ (c) \land 0 < - ℜ (- c + b__2)], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \frac{Γ (c) (\int_{0}^{1} u^{ρ - 1} {(1 - u)}^{c - ρ - 1} {}_{2}F_{1} (a__1, b__1; ρ; z__1 u) {}_{2}F_{1} (a__2, b__2; c - ρ; - (u - 1) z__2) &DifferentialD; u)}{Γ (ρ) Γ (c - ρ)}, 0 < ℜ (c)], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \frac{Γ (c) (\int_{0}^{1} \int_{0}^{1 - v} \frac{u^{- 1 + b__1} v^{b__2 - 1}}{{(1 - u - v)}^{- c + b__1 + b__2 + 1} {(- z__1 u + 1)}^{a__1} {(- v z__2 + 1)}^{a__2}} &DifferentialD; u &DifferentialD; v)}{Γ (b__1) Γ (b__2) Γ (c - b__1 - b__2)}, 0 < ℜ (b__1) \land 0 < ℜ (b__2) \land 0 < - ℜ (- c + b__1 + b__2)]$

(4)

These integral representations are also the starting point for the derivation of many of the identities known for AppellF3.

AppellF3 also satisfies a linear system of partial differential equations of second order

>	$FunctionAdvisor (DE, AppellF3)$

$[f (a__2, b__1, b__2, c, z__1, z__2) = F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2), [\frac{\partial^{2}}{\partial {z__1}^{2}} f (a__2, b__1, b__2, c, z__1, z__2) = \frac{z__2 (\frac{\partial^{2}}{\partial z__1 \partial z__2} f (a__2, b__1, b__2, c, z__1, z__2))}{z__1 (z__1 - 1)} + \frac{((- a__1 - b__1 - 1) z__1 + c) (\frac{\partial}{\partial z__1} f (a__2, b__1, b__2, c, z__1, z__2))}{z__1 (z__1 - 1)} - \frac{a__1 b__1 f (a__2, b__1, b__2, c, z__1, z__2)}{z__1 (z__1 - 1)}, \frac{\partial^{2}}{\partial z__1 \partial z__2} f (a__2, b__1, b__2, c, z__1, z__2) = \frac{z__2 (z__2 - 1) (\frac{\partial^{2}}{\partial {z__2}^{2}} f (a__2, b__1, b__2, c, z__1, z__2))}{z__1} + \frac{((a__2 + b__2 + 1) z__2 - c) (\frac{\partial}{\partial z__2} f (a__2, b__1, b__2, c, z__1, z__2))}{z__1} + \frac{a__2 b__2 f (a__2, b__1, b__2, c, z__1, z__2)}{z__1}]]$

(5)

Examples

Initialization: Set the display of special functions in output to typeset mathematical notation (textbook notation):

>	$Typesetting :- EnableTypesetRule (Typesetting :- SpecialFunctionRules) &colon;$

The conditions for both the singular and the polynomial cases can also be seen from the AppellF3. For example, the twelve polynomial cases of AppellF3 are

>	$AppellF3 :- SpecialValues :- Polynomial ()$

$12, (a1, a2, b1, b2, c, z1, z2) \mapsto [[a1 :: ℤ^{(0, -)}, a2 :: ℤ^{(0, -)}, c :: (\neg ℤ^{(0, -)})], [a1 :: ℤ^{(0, -)}, b2 :: ℤ^{(0, -)}, c :: (\neg ℤ^{(0, -)})], [b1 :: ℤ^{(0, -)}, a2 :: ℤ^{(0, -)}, c :: (\neg ℤ^{(0, -)})], [b1 :: ℤ^{(0, -)}, b2 :: ℤ^{(0, -)}, c :: (\neg ℤ^{(0, -)})], [a1 :: ℤ^{(0, -)}, a2 :: ℤ^{(0, -)}, c :: ℤ^{(0, -)}, c \leq a1 + a2], [a1 :: ℤ^{(0, -)}, b2 :: ℤ^{(0, -)}, c :: ℤ^{(0, -)}, c \leq a1 + b2], [b1 :: ℤ^{(0, -)}, a2 :: ℤ^{(0, -)}, c :: ℤ^{(0, -)}, c \leq b1 + a2], [b1 :: ℤ^{(0, -)}, b2 :: ℤ^{(0, -)}, c :: ℤ^{(0, -)}, c \leq b1 + b2], [a1 :: ℤ^{(0, -)}, c :: (\neg ℤ^{(0, -)})], [a2 :: ℤ^{(0, -)}, c :: (\neg ℤ^{(0, -)})], [b1 :: ℤ^{(0, -)}, c :: (\neg ℤ^{(0, -)})], [b2 :: ℤ^{(0, -)}, c :: (\neg ℤ^{(0, -)})]]$

(6)

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

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

>	$(%AppellF3 = AppellF3) (a__1, a__2, b__1, b__2, c, 1, z__2)$

$F_{3} (a__1, a__2, b__1, b__2, c, 1, z__2) = {}_{2}F_{1} (a__1, b__1; c; 1) {}_{3}F_{2} (a__2, b__2, c - a__1 - b__1; c - b__1, c - a__1; z__2)$

(7)

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

To see all the hypergeometric cases, enter

>	$FunctionAdvisor (specialize, AppellF3, hypergeom)$

$[F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = {}_{2}F_{1} (a__2, b__2; c; z__2), z__1 = 0 \lor a__1 = 0 \lor b__1 = 0], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = {}_{2}F_{1} (a__1, b__1; c; z__1), z__2 = 0 \lor a__2 = 0 \lor b__2 = 0], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = {}_{2}F_{1} (a__1, b__1; c; 1) {}_{3}F_{2} (a__2, b__2, c - a__1 - b__1; c - b__1, c - a__1; z__2), z__1 = 1], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = {}_{2}F_{1} (a__2, b__2; c; 1) {}_{3}F_{2} (a__1, b__1, c - a__2 - b__2; c - b__2, c - a__2; z__1), z__2 = 1], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \frac{- z__1 {}_{2}F_{1} (1, a__1; a__1 + a__2; z__1) - z__2 {}_{2}F_{1} (1, a__2; a__1 + a__2; z__2)}{(z__2 - 1) z__1 - z__2}, b__1 = 1 \land b__2 = 1 \land c = a__1 + a__2 \land - z__1 z__2 + z__1 + z__2 \neq 0], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \frac{- z__1 {}_{2}F_{1} (1, b__1; b__1 + a__2; z__1) - z__2 {}_{2}F_{1} (1, a__2; b__1 + a__2; z__2)}{(z__2 - 1) z__1 - z__2}, a__1 = 1 \land b__2 = 1 \land c = b__1 + a__2 \land - z__1 z__2 + z__1 + z__2 \neq 0], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \frac{- z__1 {}_{2}F_{1} (1, a__1; a__1 + b__2; z__1) - z__2 {}_{2}F_{1} (1, b__2; a__1 + b__2; z__2)}{(z__2 - 1) z__1 - z__2}, b__1 = 1 \land a__2 = 1 \land c = a__1 + b__2 \land - z__1 z__2 + z__1 + z__2 \neq 0], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = {(1 - z__1)}^{b__2} {}_{2}F_{1} (a__1 + b__2, b__1 + b__2; a__1 + a__2 + b__1 + b__2; z__1), (c = a__1 + a__2 + b__1 + b__2 \land z__1 \neq 1 \land z__2 = \frac{z__1}{z__1 - 1}) \lor (c = a__1 + a__2 + b__1 + b__2 \land z__2 \neq 1 \land z__1 = \frac{z__2}{z__2 - 1})], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \frac{{(1 - z__1)}^{c - 1} {}_{2}F_{1} (\frac{c}{2} + \frac{a__1}{2} - \frac{a__2}{2}, \frac{c}{2} - \frac{a__1}{2} - \frac{a__2}{2} + \frac{1}{2}; c; - 4 z__1 (z__1 - 1))}{{(1 - 2 z__1)}^{- 1 + a__2}}, (z__2 = \frac{z__1}{2 z__1 - 1} \land 2 z__1 \neq 1 \land z__1 \neq 1 \land b__1 = 1 - a__1 \land b__2 = 1 - a__2) \lor (z__1 = \frac{z__2}{2 z__2 - 1} \land 2 z__2 \neq 1 \land z__2 \neq 1 \land b__1 = 1 - a__1 \land b__2 = 1 - a__2) \lor (z__2 = \frac{z__1}{2 z__1 - 1} \land 2 z__1 \neq 1 \land z__1 \neq 1 \land a__1 = 1 - b__1 \land b__2 = 1 - a__2) \lor (z__2 = \frac{z__1}{2 z__1 - 1} \land 2 z__1 \neq 1 \land z__1 \neq 1 \land b__1 = 1 - a__1 \land a__2 = 1 - b__2)]$

(8)

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

By requesting the sum form of AppellF3, besides its double power series definition, we also see the particular form the series takes when one of the summations is performed and the result expressed in terms of 2F1 hypergeometric functions:

>	$FunctionAdvisor (sum_form, AppellF3)$

$[F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \frac{{(a__1)}_{m} {(a__2)}_{n} {(b__1)}_{m} {(b__2)}_{n} {z__1}^{m} {z__2}^{n}}{{(c)}_{m + n} m! n!}, |z__1| < 1 \land |z__2| < 1], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \sum_{k = 0}^{\infty} \frac{{(a__1)}_{k} {(b__1)}_{k} {}_{2}F_{1} (a__2, b__2; c + k; z__2) {z__1}^{k}}{{(c)}_{k} k!}, |z__1| < 1], [F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \sum_{k = 0}^{\infty} \frac{{(a__2)}_{k} {(b__2)}_{k} {}_{2}F_{1} (a__1, b__1; c + k; z__1) {z__2}^{k}}{{(c)}_{k} k!}, |z__2| < 1]$

(9)

As indicated in the formulas above, for AppellF3 (also for AppellF1) the domain of convergence of the single sum with hypergeometric coefficients is larger than the domain of convergence of the double series, because the hypergeometric coefficient in the single sum - say the one in $z_{2}$ - analytically extends the series with regards to the other variable - say $z_{1}$ - entering the hypergeometric coefficient. Hence, for AppellF3 (also for AppellF1), the case where one of the two variables, $z_{1}$ or $z_{2}$ , is equal to 1, is convergent only when the corresponding hypergeometric coefficient in the single sum form is convergent. For instance, the convergent case at $z_{1} = 1$ requires that $0 < Re (c - a_{1} - b_{1})$ .

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

A contiguity transformation for AppellF3

>	$AppellF3 (a__1, a__2, b__1, b__2, c, z__1, z__2) = AppellF3 :- Transformations [Contiguity] [1] (a__1, a__2, b__1, b__2, c, z__1, z__2)$

$F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = [F_{3} (a__1, a__2, b__1, b__2 + n, c, z__1, z__2) - (\sum_{k = 1}^{n} \frac{(\binom{n}{k}) {(a__2)}_{k} {z__2}^{k} F_{3} (a__1, b__2 + k, b__1, k + a__2, k + c, z__1, z__2)}{{(c)}_{k}}), z__2 \neq 1 \land (c :: (\neg ℤ^{(0, -)}) \lor (a__2 :: ℤ^{(0, -)} \land c < a__2) \lor n \leq |c|)]$

(10)

The contiguity transformations available in this way are

>	$indices (AppellF3 :- Transformations [Contiguity])$

$[1], [2], [3], [4], [5], [6]$

(11)

By using differential algebra techniques, the PDE system satisfied by AppellF3 can be transformed into an equivalent PDE system where one of the equations is a linear ODE in $z_{2}$ parametrized by $z_{1}$ . In the case of AppellF3 this linear ODE is of fourth order and can be computed as follows

>	$F3 (z__1, z__2) = AppellF3 (a__1, a__2, b__1, b__2, c, z__1, z__2)$

$F3 (z__1, z__2) = F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2)$

(12)

>	$simplify (op ([1, 2], PDEtools :- casesplit (PDEtools :- dpolyform (, no_Fn), lex)))$

$\frac{\partial^{4}}{\partial {z__2}^{4}} F3 (z__1, z__2) = \frac{2 z__2 (- (z__1 - 1) (a__2 + b__2 + 4) {z__2}^{2} + ((c - \frac{a__1}{2} + a__2 - \frac{b__1}{2} + b__2 + \frac{11}{2}) z__1 - \frac{c}{2} - \frac{a__2}{2} - \frac{b__2}{2} - \frac{5}{2}) z__2 - (c - \frac{a__1}{2} - \frac{b__1}{2} + \frac{3}{2}) z__1) (\frac{\partial^{3}}{\partial {z__2}^{3}} F3 (z__1, z__2)) + (- ({a__2}^{2} + (4 b__2 + 9) a__2 + {b__2}^{2} + 9 b__2 + 14) (z__1 - 1) {z__2}^{2} + (((2 c - a__1 - b__1 + 2 b__2 + 5) a__2 + (2 c - a__1 - b__1 + 5) b__2 + 6 c - 3 a__1 - 3 b__1 + 11) z__1 + (- c - b__2 - 2) a__2 + (- c - 2) b__2 - 3 c - 4) z__2 - z__1 (c - b__1 + 1) (c - a__1 + 1)) (\frac{\partial^{2}}{\partial {z__2}^{2}} F3 (z__1, z__2)) + 2 ((- (z__1 - 1) (a__2 + b__2 + 2) z__2 + (c - \frac{a__1}{2} - \frac{b__1}{2} + \frac{1}{2}) z__1 - \frac{c}{2}) (\frac{\partial}{\partial z__2} F3 (z__1, z__2)) - \frac{a__2 b__2 F3 (z__1, z__2) (z__1 - 1)}{2}) (b__2 + 1) (a__2 + 1)}{{z__2}^{2} (z__2 - 1) ((z__1 - 1) z__2 - z__1)}$

(13)

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

>	$DEtools [singularities] (subs (F3 (z__1, z__2) = F3 (z__2),))$

$regular = \{0, 1, \infty, \frac{z__1}{z__1 - 1}\}, irregular = \emptyset$

(14)

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

>	$FunctionAdvisor (AppellF3)$

AppellF3

describe

$AppellF3 = Appell 2-variable hypergeometric function F3$

definition

$F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2) = \sum_{_k1 = 0}^{\infty} \sum_{_k2 = 0}^{\infty} \frac{{(a__1)}_{_k1} {(a__2)}_{_k2} {(b__1)}_{_k1} {(b__2)}_{_k2} {z__1}^{_k1} {z__2}^{_k2}}{{(c)}_{_k1 +_k2}_k1!_k2!}$

$|z__1| < 1 \land |z__2| < 1$

classify function

$Appell$

symmetries

$F_{3} (a__2, a__1, b__2, b__1, c, z__2, z__1) = F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2)$

$F_{3} (b__1, a__2, a__1, b__2, c, z__1, z__2) = F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2)$

$F_{3} (a__1, b__2, b__1, a__2, c, z__1, z__2) = F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2)$

plot

singularities

$F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2)$

$(c :: ℤ^{(0, -)} \land a__1 :: (\neg ℤ^{(0, -)}) \land b__1 :: (\neg ℤ^{(0, -)})) \lor (c :: ℤ^{(0, -)} \land a__2 :: (\neg ℤ^{(0, -)}) \land b__2 :: (\neg ℤ^{(0, -)})) \lor (c :: ℤ^{(0, -)} \land a__1 :: (\neg ℤ^{(0, -)}) \land a__2 :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land b__2 :: (\neg ℤ^{(0, -)}) \land b__1 + a__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: (\neg ℤ^{(0, -)}) \land a__2 :: (\neg ℤ^{(0, -)}) \land b__1 :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land b__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: (\neg ℤ^{(0, -)}) \land a__2 :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land b__1 + a__2 < c \land b__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: ℤ^{(0, -)} \land b__1 :: (\neg ℤ^{(0, -)}) \land b__2 :: (\neg ℤ^{(0, -)}) \land a__1 + a__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: (\neg ℤ^{(0, -)}) \land b__1 :: (\neg ℤ^{(0, -)}) \land b__2 :: ℤ^{(0, -)} \land a__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: ℤ^{(0, -)} \land b__1 :: (\neg ℤ^{(0, -)}) \land b__2 :: ℤ^{(0, -)} \land a__1 + a__2 < c \land a__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land a__1 + a__2 < c \land a__1 + b__2 < c \land b__1 + a__2 < c \land b__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: (\neg ℤ^{(0, -)}) \land b__1 :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land a__1 + b__2 < c \land b__1 + b__2 < c) \lor (c :: ℤ^{(0, -)} \land a__1 :: ℤ^{(0, -)} \land a__2 :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land b__2 :: (\neg ℤ^{(0, -)}) \land a__1 + a__2 < c \land b__1 + a__2 < c)$

branch points

$F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2)$

$(a__1 :: (\neg ℤ^{(0, -)}) \land b__1 :: (\neg ℤ^{(0, -)}) \land z__1 \in [1, \infty + \infty I]) \lor (a__2 :: (\neg ℤ^{(0, -)}) \land b__2 :: (\neg ℤ^{(0, -)}) \land z__2 \in [1, \infty + \infty I])$

branch cuts

$F_{3} (a__1, a__2, b__1, b__2, c, z__1, z__2)$

$(a__1 :: (\neg ℤ^{(0, -)}) \land b__1 :: (\neg ℤ^{(0, -)}) \land 1 < z__1) \lor (a__2 :: (\neg ℤ^{(0, -)}) \land b__2 :: (\neg ℤ^{(0, -)}) \land 1 < z__2)$

special values