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AppellF2

The AppellF2 function

	Calling Sequence
	AppellF2( $a, b_{1}, b_{2}, c_{1}, c_{2}, z_{1}, z_{2}$ )

Parameters

$a$	-	algebraic expression
$b_{1}$	-	algebraic expression
$b_{2}$	-	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, AppellF2, 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, AppellF2 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 AppellF2 series and the corresponding domain of convergence can be seen through the FunctionAdvisor

>	$FunctionAdvisor (definition, AppellF2)$

$[F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \sum_{_k1 = 0}^{\infty} \sum_{_k2 = 0}^{\infty} \frac{{(a)}_{_k1 +_k2} {(b__1)}_{_k1} {(b__2)}_{_k2} {z__1}^{_k1} {z__2}^{_k2}}{{(c__1)}_{_k1} {(c__2)}_{_k2}_k1!_k2!}, |z__1| + |z__2| < 1]$

(1)

A distinction is made between the AppellF2 doubly hypergeometric series, with the restricted domain of convergence shown above, and the AppellF2 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 AppellF2 variables subscripted with the numbers 1 and 2, the function remains the same; hence

>	$FunctionAdvisor (symmetries, AppellF2)$

$[F_{2} (a, b__2, b__1, c__2, c__1, z__2, z__1) = F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2)]$

(2)

From the series' definition, AppellF2 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 AppellF2 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, AppellF2 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 either the following conditions hold

>	$FunctionAdvisor (singularities, AppellF2)$

$[F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2), (c__1 :: ℤ^{(0, -)} \land a :: (\neg ℤ^{(0, -)}) \land b__1 :: (\neg ℤ^{(0, -)})) \lor (c__1 :: ℤ^{(0, -)} \land a :: ℤ^{(0, -)} \land b__1 :: (\neg ℤ^{(0, -)}) \land a < c__1) \lor (c__1 :: ℤ^{(0, -)} \land a :: (\neg ℤ^{(0, -)}) \land b__1 :: ℤ^{(0, -)} \land b__1 < c__1) \lor (c__1 :: ℤ^{(0, -)} \land a :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land a < c__1 \land b__1 < c__1) \lor (c__2 :: ℤ^{(0, -)} \land a :: (\neg ℤ^{(0, -)}) \land b__2 :: (\neg ℤ^{(0, -)})) \lor (c__2 :: ℤ^{(0, -)} \land a :: ℤ^{(0, -)} \land b__2 :: (\neg ℤ^{(0, -)}) \land a < c__2) \lor (c__2 :: ℤ^{(0, -)} \land a :: (\neg ℤ^{(0, -)}) \land b__2 :: ℤ^{(0, -)} \land b__2 < c__2) \lor (c__2 :: ℤ^{(0, -)} \land a :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land a < c__2 \land b__2 < c__2)]$

(3)

The AppellF2 series is analytically extended to the AppellF2 function defined over the whole complex plane using identities and mainly by integral representations in terms of Eulerian integrals:

>	$FunctionAdvisor (integral_form, AppellF2)$

$[F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{Γ (c__1) (\int_{0}^{1} \frac{u^{b__1 - 1} {}_{2}F_{1} (a, b__2; c__2; - \frac{z__2}{u z__1 - 1})}{{(1 - u)}^{- c__1 + b__1 + 1} {(- u z__1 + 1)}^{a}} &DifferentialD; u)}{Γ (b__1) Γ (c__1 - b__1)}, z__1 \neq 1 \land 0 < ℜ (b__1) \land 0 < - ℜ (- c__1 + b__1)], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{Γ (c__2) (\int_{0}^{1} \frac{u^{b__2 - 1} {}_{2}F_{1} (a, b__1; c__1; - \frac{z__1}{z__2 u - 1})}{{(1 - u)}^{1 + b__2 - c__2} {(- z__2 u + 1)}^{a}} &DifferentialD; u)}{Γ (b__2) Γ (c__2 - b__2)}, z__2 \neq 1 \land 0 < ℜ (b__2) \land 0 < - ℜ (- c__2 + b__2)], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{Γ (c__1) Γ (c__2) (\int_{0}^{1} \int_{0}^{1} \frac{u^{b__1 - 1} v^{b__2 - 1}}{{(1 - u)}^{- c__1 + b__1 + 1} {(1 - v)}^{1 + b__2 - c__2} {(- u z__1 - v z__2 + 1)}^{a}} &DifferentialD; u &DifferentialD; v)}{Γ (b__1) Γ (b__2) Γ (c__1 - b__1) Γ (c__2 - b__2)}, 0 < ℜ (b__1) \land 0 < ℜ (b__2) \land 0 < - ℜ (- c__1 + b__1) \land 0 < - ℜ (- c__2 + b__2)], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{\int_{0}^{\infty} \frac{u^{a - 1} {}_{1}F_{1} (b__1; c__1; u z__1) {}_{1}F_{1} (b__2; c__2; z__2 u)}{{&ExponentialE;}^{u}} &DifferentialD; u}{Γ (a)}, ℜ (z__1 + z__2) < 1 \land 0 < ℜ (a)]$

(4)

These integral representations are also the starting point for the derivation of many of the identities known for AppellF2.

AppellF2 also satisfies a linear system of partial differential equations of second order

>	$FunctionAdvisor (DE, AppellF2)$

$[f (a, b__1, b__2, c__1, c__2, z__1, z__2) = F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2), [\frac{\partial^{2}}{\partial {z__1}^{2}} f (a, b__1, b__2, c__1, c__2, z__1, z__2) = - \frac{z__2 (\frac{\partial^{2}}{\partial z__1 \partial z__2} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1 - 1} + \frac{((- a - b__1 - 1) z__1 + c__1) (\frac{\partial}{\partial z__1} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1 (z__1 - 1)} - \frac{b__1 z__2 (\frac{\partial}{\partial z__2} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1 (z__1 - 1)} - \frac{f (a, b__1, b__2, c__1, c__2, z__1, z__2) a b__1}{z__1 (z__1 - 1)}, \frac{\partial^{2}}{\partial z__1 \partial z__2} f (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{(1 - z__2) (\frac{\partial^{2}}{\partial {z__2}^{2}} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1} - \frac{b__2 (\frac{\partial}{\partial z__1} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__2} + \frac{((- a - b__2 - 1) z__2 + c__2) (\frac{\partial}{\partial z__2} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1 z__2} - \frac{f (a, b__1, b__2, c__1, c__2, z__1, z__2) a b__2}{z__1 z__2}]]$

(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 AppellF2. For example, the fourteen polynomial cases of AppellF2 are

>	$AppellF2 :- SpecialValues :- Polynomial ()$

$14, (a, b1, b2, c1, c2, z1, z2) \mapsto'[[a :: ℤ^{(0, -)}, c1 :: (\neg ℤ^{(0, -)}), c2 :: (\neg ℤ^{(0, -)})], [a :: ℤ^{(0, -)}, c1 :: ℤ^{(0, -)}, c1 \leq a, c2 :: (\neg ℤ^{(0, -)})], [a :: ℤ^{(0, -)}, c1 :: (\neg ℤ^{(0, -)}), c2 :: ℤ^{(0, -)}, c2 \leq a], [a :: ℤ^{(0, -)}, c1 :: ℤ^{(0, -)}, c2 :: ℤ^{(0, -)}, c1 \leq a, c2 \leq a], [b1 :: ℤ^{(0, -)}, b2 :: ℤ^{(0, -)}, c1 :: (\neg ℤ^{(0, -)}), c2 :: (\neg ℤ^{(0, -)})], [b1 :: ℤ^{(0, -)}, b2 :: ℤ^{(0, -)}, c1 :: ℤ^{(0, -)}, c1 \leq b1, c2 :: (\neg ℤ^{(0, -)})], [b1 :: ℤ^{(0, -)}, b2 :: ℤ^{(0, -)}, c1 :: (\neg ℤ^{(0, -)}), c2 :: ℤ^{(0, -)}, c2 \leq b2], [b1 :: ℤ^{(0, -)}, b2 :: ℤ^{(0, -)}, c1 :: ℤ^{(0, -)}, c1 \leq b1, c2 :: ℤ^{(0, -)}, c2 \leq b2], [b1 :: ℤ^{(0, -)}, c1 :: (\neg ℤ^{(0, -)}), c2 :: (\neg ℤ^{(0, -)})], [b1 :: ℤ^{(0, -)}, c1 :: ℤ^{(0, -)}, c1 \leq b1, c2 :: (\neg ℤ^{(0, -)})], [b2 :: ℤ^{(0, -)}, c1 :: (\neg ℤ^{(0, -)}), c2 :: (\neg ℤ^{(0, -)})], [b2 :: ℤ^{(0, -)}, c1 :: (\neg ℤ^{(0, -)}), c2 :: ℤ^{(0, -)}, c2 \leq b2], [a :: ℤ^{(0, -)}, b1 :: ℤ^{(0, -)}, c1 :: ℤ^{(0, -)}, a < b1, c1 \leq b1, c2 :: ℤ^{(0, -)}, c2 \leq a], [a :: ℤ^{(0, -)}, b2 :: ℤ^{(0, -)}, c1 :: ℤ^{(0, -)}, c1 \leq a, c2 :: ℤ^{(0, -)}, a < b2, c2 \leq b2]]'$

(6)

Likewise, the conditions for the singular cases of AppellF2 can be seen either using the FunctionAdvisor or entering AppellF2:-Singularities(), so with no arguments.

For particular values of its parameters, AppellF2 is related to the hypergeometric function. These hypergeometric cases are returned automatically. For example, for $c_{1} = b_{1}$ ,

>	$(%AppellF2 = AppellF2) (a, b__1, b__2, b__1, c__2, z__1, z__2)$

$F_{2} (a, b__1, b__2, b__1, c__2, z__1, z__2) = {(1 - z__1)}^{- a} {}_{2}F_{1} (a, b__2; c__2; \frac{z__2}{1 - z__1})$

(7)

To see all the hypergeometric cases, enter

>	$FunctionAdvisor (specialize, AppellF2, hypergeom)$

$[F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{2}F_{1} (a, b__2; c__2; z__2), z__1 = 0 \lor b__1 = 0], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{2}F_{1} (a, b__1; c__1; z__1), z__2 = 0 \lor b__2 = 0], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{{}_{2}F_{1} (a, b__2; c__2; \frac{z__2}{1 - z__1})}{{(1 - z__1)}^{a}}, c__1 = b__1 \land z__1 \neq 1], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{{}_{2}F_{1} (a, b__1; c__1; \frac{z__1}{1 - z__2})}{{(1 - z__2)}^{a}}, c__2 = b__2 \land z__2 \neq 1], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{4}F_{3} (\frac{a}{2}, \frac{1}{2} + \frac{a}{2}, \frac{b__2}{2} + \frac{b__1}{2}, \frac{b__2}{2} + \frac{b__1}{2} + \frac{1}{2}; \frac{1}{2} + b__1, b__2 + \frac{1}{2}, b__2 + b__1; {z__1}^{2}), c__1 = 2 b__1 \land c__2 = 2 b__2 \land z__2 = - z__1], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{4}F_{3} (b__1, \frac{a}{2}, c__1 - b__1, \frac{1}{2} + \frac{a}{2}; c__1, \frac{c__1}{2}, \frac{c__1}{2} + \frac{1}{2}; {z__1}^{2}), b__2 = b__1 \land c__2 = c__1 \land z__2 = - z__1], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{4}F_{3} (\frac{a}{2}, \frac{1}{2} + \frac{a}{2}, \frac{c__1}{2} + \frac{1}{2} - b__1, \frac{1}{2} - \frac{c__1}{2} + b__1; \frac{1}{2}, \frac{c__1}{2} + \frac{1}{2}, \frac{3}{2} - \frac{c__1}{2}; {z__1}^{2}) - \frac{a (1 - c__1) (c__1 - 2 b__1) z__1 {}_{4}F_{3} (\frac{a}{2} + 1, \frac{1}{2} + \frac{a}{2}, 1 - \frac{c__1}{2} + b__1, 1 + \frac{c__1}{2} - b__1; \frac{3}{2}, 2 - \frac{c__1}{2}, 1 + \frac{c__1}{2}; {z__1}^{2})}{c__1 (2 - c__1)}, (b__2 = 1 + b__1 - c__1 \land c__2 = 2 - c__1 \land z__2 = - z__1 \land c__1 \neq 2) \lor (b__1 = 1 + b__2 - c__2 \land c__1 = 2 - c__2 \land z__2 = - z__1 \land c__2 \neq 2)]$

(8)

Other special values of AppellF2 can be seen using FunctionAdvisor(special_values, AppellF2).

By requesting the sum form of AppellF2, 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:

>	$FunctionAdvisor (sum_form, AppellF2)$

$[F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \frac{{(a)}_{m + n} {(b__1)}_{m} {(b__2)}_{n} {z__1}^{m} {z__2}^{n}}{{(c__1)}_{m} {(c__2)}_{n} m! n!}, |z__2| + |z__1| < 1], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \sum_{k = 0}^{\infty} \frac{{(a)}_{k} {(b__1)}_{k} {}_{2}F_{1} (a + k, b__2; c__2; z__2) {z__1}^{k}}{{(c__1)}_{k} k!}, |z__2| + |z__1| < 1], [F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \sum_{k = 0}^{\infty} \frac{{(a)}_{k} {(b__2)}_{k} {}_{2}F_{1} (a + k, b__1; c__1; z__1) {z__2}^{k}}{{(c__2)}_{k} k!}, |z__2| + |z__1| < 1]$

(9)

As indicated in the formulas above, for AppellF2 (also for AppellF4), 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.

AppellF2 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,

>	$AppellF2 (a, b__1, b__2, c__1, c__2, z__1, z__2) = AppellF2 :- Transformations [Euler] [1] (a, b__1, b__2, c__1, c__2, z__1, z__2)$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{F_{2} (a, c__1 - b__1, b__2, c__1, c__2, \frac{z__1}{z__1 - 1}, \frac{z__2}{1 - z__1})}{{(1 - z__1)}^{a}}$

(10)

Among other situations, this identity is useful when the sum 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 AppellF2 on the right-hand side is smaller than 1. On the other hand, unlike the case of the other three Appell functions, none of the two Euler type transformations or hypergeometric special cases of AppellF2 are of help to analytically extend to the whole complex plane the AppellF2 series when either $z_{1} = 1$ or $z_{2} = 1$ .

A contiguity transformation for AppellF2

>	$AppellF2 (a, b__1, b__2, c__1, c__2, z__1, z__2) = AppellF2 :- Transformations [Contiguity] [1] (a, b__1, b__2, c__1, c__2, z__1, z__2)$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{- a F_{2} (a + 1, b__1, b__2, c__1, c__2, z__1, z__2) + b__1 F_{2} (a, b__1 + 1, b__2, c__1, c__2, z__1, z__2) + b__2 F_{2} (a, b__1, b__2 + 1, c__1, c__2, z__1, z__2)}{- a + b__1 + b__2}$

(11)

The contiguity transformations available in this way are

>	$indices (AppellF2 :- Transformations [Contiguity])$

$[1], [2], [3], [4], [5], [6], [7], [8]$

(12)

By using differential algebra techniques, the PDE system satisfied by AppellF2 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 AppellF2 this linear ODE is of fourth order and can be computed as follows

>	$F2 (z__1, z__2) = AppellF2 (a, b__1, b__2, c__1, c__2, z__1, z__2)$

$F2 (z__1, z__2) = F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2)$

(13)

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

$\frac{\partial^{4}}{\partial {z__2}^{4}} F2 (z__1, z__2) = \frac{- ((2 a + 2 b__2 - c__1 + 8) {z__2}^{2} + ((z__1 - 2) b__2 + (z__1 - 2) a + (z__1 - 2) c__2 + (- b__1 + 5) z__1 + c__1 - 10) z__2 - 2 (z__1 - 1) (c__2 + 1)) z__2 (\frac{\partial^{3}}{\partial {z__2}^{3}} F2 (z__1, z__2)) + ((- {b__2}^{2} + (- 4 a + 2 c__1 - 9) b__2 - a^{2} + (c__1 - 9) a + 4 c__1 - 14) {z__2}^{2} + (((- z__1 + 2) a + (- z__1 + 2) c__2 + (b__1 - 2) z__1 - c__1 + 4) b__2 - (z__1 - 2) (c__2 + 2) a + ((b__1 - 3) z__1 - c__1 + 6) c__2 + (2 b__1 - 4) z__1 - 2 c__1 + 8) z__2 + c__2 (z__1 - 1) (c__2 + 1)) (\frac{\partial^{2}}{\partial {z__2}^{2}} F2 (z__1, z__2)) - ((((2 a - c__1 + 2) b__2 + 2 (a + 1) (a - c__1 + 2)) z__2 + c__2 ((z__1 - 2) a + (- b__1 + 1) z__1 + c__1 - 2)) (\frac{\partial}{\partial z__2} F2 (z__1, z__2)) + F2 (z__1, z__2) a b__2 (a - c__1 + 1)) (b__2 + 1)}{{z__2}^{2} (z__2 - 1) (z__2 + z__1 - 1)}$

(14)

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

>	$DEtools [singularities] (subs (F2 (z__1, z__2) = F2 (z__2),))$

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

(15)

You can also see a general presentation of AppellF2, organized into sections and including plots, using the FunctionAdvisor

>	$FunctionAdvisor (AppellF2)$

AppellF2

describe

$AppellF2 = Appell 2-variable hypergeometric function F2$

definition

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \sum_{_k1 = 0}^{\infty} \sum_{_k2 = 0}^{\infty} \frac{{(a)}_{_k1 +_k2} {(b__1)}_{_k1} {(b__2)}_{_k2} {z__1}^{_k1} {z__2}^{_k2}}{{(c__1)}_{_k1} {(c__2)}_{_k2}_k1!_k2!}$

$|z__1| + |z__2| < 1$

classify function

$Appell$

symmetries

$F_{2} (a, b__2, b__1, c__2, c__1, z__2, z__1) = F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2)$

plot

singularities

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2)$

$(c__1 :: ℤ^{(0, -)} \land a :: (\neg ℤ^{(0, -)}) \land b__1 :: (\neg ℤ^{(0, -)})) \lor (c__1 :: ℤ^{(0, -)} \land a :: ℤ^{(0, -)} \land b__1 :: (\neg ℤ^{(0, -)}) \land a < c__1) \lor (c__1 :: ℤ^{(0, -)} \land a :: (\neg ℤ^{(0, -)}) \land b__1 :: ℤ^{(0, -)} \land b__1 < c__1) \lor (c__1 :: ℤ^{(0, -)} \land a :: ℤ^{(0, -)} \land b__1 :: ℤ^{(0, -)} \land a < c__1 \land b__1 < c__1) \lor (c__2 :: ℤ^{(0, -)} \land a :: (\neg ℤ^{(0, -)}) \land b__2 :: (\neg ℤ^{(0, -)})) \lor (c__2 :: ℤ^{(0, -)} \land a :: ℤ^{(0, -)} \land b__2 :: (\neg ℤ^{(0, -)}) \land a < c__2) \lor (c__2 :: ℤ^{(0, -)} \land a :: (\neg ℤ^{(0, -)}) \land b__2 :: ℤ^{(0, -)} \land b__2 < c__2) \lor (c__2 :: ℤ^{(0, -)} \land a :: ℤ^{(0, -)} \land b__2 :: ℤ^{(0, -)} \land a < c__2 \land b__2 < c__2)$

branch points

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2)$

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

branch cuts

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2)$

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

special values

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = 1$

$z__1 = 0 \land z__2 = 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = 1$

$a = 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = 1$

$b__1 = 0 \land b__2 = 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{2}F_{1} (a, b__2; c__2; z__2)$

$z__1 = 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{2}F_{1} (a, b__1; c__1; z__1)$

$z__2 = 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{2}F_{1} (a, b__2; c__2; z__2)$

$b__1 = 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{2}F_{1} (a, b__1; c__1; z__1)$

$b__2 = 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {(1 - z__1)}^{- a} {}_{2}F_{1} (a, b__2; c__2; \frac{z__2}{1 - z__1})$

$c__1 = b__1 \land z__1 \neq 1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {(1 - z__2)}^{- a} {}_{2}F_{1} (a, b__1; c__1; \frac{z__1}{1 - z__2})$

$c__2 = b__2 \land z__2 \neq 1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{4}F_{3} (\frac{a}{2}, \frac{1}{2} + \frac{a}{2}, \frac{b__2}{2} + \frac{b__1}{2}, \frac{b__2}{2} + \frac{b__1}{2} + \frac{1}{2}; b__2 + b__1, \frac{1}{2} + b__1, b__2 + \frac{1}{2}; {z__1}^{2})$

$c__1 = 2 b__1 \land c__2 = 2 b__2 \land z__2 = - z__1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{4}F_{3} (b__1, \frac{a}{2}, c__1 - b__1, \frac{1}{2} + \frac{a}{2}; c__1, \frac{c__1}{2}, \frac{c__1}{2} + \frac{1}{2}; {z__1}^{2})$

$b__2 = b__1 \land c__2 = c__1 \land z__2 = - z__1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{4}F_{3} (\frac{a}{2}, \frac{1}{2} + \frac{a}{2}, \frac{c__1}{2} + \frac{1}{2} - b__1, \frac{1}{2} - \frac{c__1}{2} + b__1; \frac{1}{2}, \frac{c__1}{2} + \frac{1}{2}, \frac{3}{2} - \frac{c__1}{2}; {z__1}^{2}) - \frac{a (1 - c__1) (c__1 - 2 b__1) z__1 {}_{4}F_{3} (\frac{a}{2} + 1, \frac{1}{2} + \frac{a}{2}, 1 - \frac{c__1}{2} + b__1, 1 + \frac{c__1}{2} - b__1; \frac{3}{2}, 2 - \frac{c__1}{2}, 1 + \frac{c__1}{2}; {z__1}^{2})}{c__1 (2 - c__1)}$

$b__2 = 1 + b__1 - c__1 \land c__2 = 2 - c__1 \land z__2 = - z__1 \land c__1 \neq 2$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {}_{4}F_{3} (\frac{a}{2}, \frac{1}{2} + \frac{a}{2}, \frac{c__2}{2} + \frac{1}{2} - b__2, \frac{1}{2} - \frac{c__2}{2} + b__2; \frac{1}{2}, \frac{c__2}{2} + \frac{1}{2}, \frac{3}{2} - \frac{c__2}{2}; {z__2}^{2}) - \frac{a (1 - c__2) (c__2 - 2 b__2) z__2 {}_{4}F_{3} (\frac{a}{2} + 1, \frac{1}{2} + \frac{a}{2}, 1 - \frac{c__2}{2} + b__2, 1 + \frac{c__2}{2} - b__2; \frac{3}{2}, 2 - \frac{c__2}{2}, 1 + \frac{c__2}{2}; {z__2}^{2})}{c__2 (2 - c__2)}$

$b__1 = 1 + b__2 - c__2 \land c__1 = 2 - c__2 \land z__2 = - z__1 \land c__2 \neq 2$

identities

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{F_{2} (a, c__1 - b__1, b__2, c__1, c__2, \frac{z__1}{- 1 + z__1}, \frac{z__2}{1 - z__1})}{{(1 - z__1)}^{a}}$

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

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{F_{2} (a, c__1 - b__1, c__2 - b__2, c__1, c__2, \frac{z__1}{- 1 + z__1 + z__2}, \frac{z__2}{- 1 + z__1 + z__2})}{{(1 - z__1 - z__2)}^{a}}$

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

$z__2 \neq 1 \land z__1 \neq 1 \land a \neq b__2 + b__1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = - \frac{b__1 z__1 (\sum_{k = 0}^{n - 1} F_{2} (a + k + 1, b__1 + 1, b__2, c__1 + 1, c__2, z__1, z__2))}{c__1} - \frac{b__2 z__2 (\sum_{k = 0}^{n - 1} F_{2} (a + k + 1, b__1, b__2 + 1, c__1, c__2 + 1, z__1, z__2))}{c__2} + F_{2} (a + n, b__1, b__2, c__1, c__2, z__1, z__2)$

$c__1 \neq 0 \land c__2 \neq 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = F_{2} (a, b__1 + n, b__2, c__1, c__2, z__1, z__2) - \frac{a z__1 (\sum_{k = 1}^{n} F_{2} (a + 1, b__1 + k, b__2, c__1 + 1, c__2, z__1, z__2))}{c__1}$

$c__1 \neq 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = F_{2} (a, b__1, b__2, c__1 - n, c__2, z__1, z__2) - a b__1 z__1 (\sum_{k = 1}^{n} \frac{F_{2} (a + 1, b__1 + 1, b__2, c__1 - k + 2, c__2, z__1, z__2)}{(c__1 - k) (c__1 - k + 1)})$

$c__1 :: (\neg ℤ^{+}) \lor n < c__1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{{(−1)}^{n} {(a)}_{n} {(b__2)}_{n} {z__2}^{n} F_{2} (a + n, b__1, n + b__2, c__1, n + c__2, z__1, z__2)}{{(c__2)}_{n} {(c__2 - 1)}_{n}} - (\sum_{k = 1}^{n} \frac{{(−1)}^{k} (\binom{n}{k}) {(1 - c__2)}_{k} F_{2} (a, b__1, b__2, c__1, c__2 - k, z__1, z__2)}{{(2 - c__2 - n)}_{k}})$

$c__2 :: (\neg ℤ) \lor n < c__2$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{b__1 F_{2} (a, b__1 + 1, b__2, c__1, c__2, z__1, z__2)}{1 + b__1 - c__1} - \frac{F_{2} (a, b__1, b__2, - 1 + c__1, c__2, z__1, z__2) (- 1 + c__1)}{1 + b__1 - c__1}$

$c__1 \neq 1 \land 1 + b__1 - c__1 \neq 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{F_{2} (a, b__1 + 1, b__2, c__1 + 1, c__2, z__1, z__2) b__1}{c__1} - \frac{(b__1 - c__1) F_{2} (a, b__1, b__2, c__1 + 1, c__2, z__1, z__2)}{c__1}$

$c__1 \neq 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = F_{2} (a, b__1, b__2, c__1 + 1, c__2, z__1, z__2) + \frac{a z__1 F_{2} (a + 1, b__1, b__2, c__1 + 1, c__2, z__1, z__2)}{c__1} + \frac{F_{2} (a + 1, b__1, b__2, c__1 + 2, c__2, z__1, z__2) a (b__1 - c__1 - 1) z__1}{c__1 (c__1 + 1)}$

$c__1 \neq 0 \land c__1 \neq −1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{F_{1} (b__1, - b__2 + a, b__2, c__1, z__1, \frac{z__1}{1 - z__2})}{{(1 - z__2)}^{b__2}}$

$c__2 = a \land z__2 \neq 1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{F_{1} (b__2, a - b__1, b__1, c__2, z__2, \frac{z__2}{1 - z__1})}{{(1 - z__1)}^{b__1}}$

$c__1 = a \land z__1 \neq 1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{{(\frac{- 1 + z__2}{- 1 + z__1 + z__2})}^{b__2} F_{3} (b__1, b__2, - b__2 + a, c__1 - b__1, c__1, z__1, \frac{z__1}{- 1 + z__1 + z__2})}{{(1 - z__2)}^{b__2}}$

$c__2 = a \land z__2 \neq 1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{{(\frac{- 1 + z__1}{- 1 + z__1 + z__2})}^{b__1} F_{3} (b__2, b__1, a - b__1, c__2 - b__2, c__2, z__2, \frac{z__2}{- 1 + z__1 + z__2})}{{(1 - z__1)}^{b__1}}$

$c__1 = a \land z__1 \neq 1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = F_{4} (a, b__1, c__1, c__2, z__1 (1 - z__2), z__2 (1 - z__1))$

$a = c__1 + c__2 - 1 \land b__1 = b__2$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{F_{4} (\frac{a}{2}, \frac{1}{2} + \frac{a}{2}, \frac{1}{2} + b__1, b__2 + \frac{1}{2}, \frac{{z__1}^{2}}{{(- 2 + z__1 + z__2)}^{2}}, \frac{{z__2}^{2}}{{(- 2 + z__1 + z__2)}^{2}})}{{(1 - \frac{z__1}{2} - \frac{z__2}{2})}^{a}}$

$c__1 = 2 b__1 \land c__2 = 2 b__2 \land 1 - \frac{z__1}{2} - \frac{z__2}{2} \neq 0$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {(1 + \sqrt{\frac{(- 4 z__2 + 8) \sqrt{1 - z__2} + {z__2}^{2} - 8 z__2 + 8}{{z__2}^{2}}})}^{2 a} F_{4} (a, b__1, a - b__1 + 1, c__1, \frac{(- 4 z__2 + 8) \sqrt{1 - z__2} + {z__2}^{2} - 8 z__2 + 8}{{z__2}^{2}}, \frac{2 z__1 (\sqrt{\frac{(- 4 z__2 + 8) \sqrt{1 - z__2} + {z__2}^{2} - 8 z__2 + 8}{{z__2}^{2}}} {z__2}^{2} + (z__2 - 2) (z__2 - 2 \sqrt{1 - z__2} - 2))}{{z__2}^{2}})$

$b__2 = a - b__1 + \frac{1}{2} \land c__2 = 2 b__2 \land {(ℜ (z__2) - 1)}^{2} + {ℑ (z__2)}^{2} \leq 1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = {(1 + \sqrt{\frac{(- 4 z__1 + 8) \sqrt{1 - z__1} + {z__1}^{2} - 8 z__1 + 8}{{z__1}^{2}}})}^{2 a} F_{4} (a, b__2, a - b__2 + 1, c__2, \frac{(- 4 z__1 + 8) \sqrt{1 - z__1} + {z__1}^{2} - 8 z__1 + 8}{{z__1}^{2}}, \frac{2 z__2 (\sqrt{\frac{(- 4 z__1 + 8) \sqrt{1 - z__1} + {z__1}^{2} - 8 z__1 + 8}{{z__1}^{2}}} {z__1}^{2} + (z__1 - 2) (z__1 - 2 \sqrt{1 - z__1} - 2))}{{z__1}^{2}})$

$b__1 = a - b__2 + \frac{1}{2} \land c__1 = 2 b__1 \land {(ℜ (z__1) - 1)}^{2} + {ℑ (z__1)}^{2} \leq 1$

sum form

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \sum_{m = 0}^{\infty} \sum_{n = 0}^{\infty} \frac{{(a)}_{m + n} {(b__1)}_{m} {(b__2)}_{n} {z__1}^{m} {z__2}^{n}}{{(c__1)}_{m} {(c__2)}_{n} m! n!}$

$|z__2| + |z__1| < 1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \sum_{k = 0}^{\infty} \frac{{(a)}_{k} {(b__1)}_{k} {}_{2}F_{1} (a + k, b__2; c__2; z__2) {z__1}^{k}}{{(c__1)}_{k} k!}$

$|z__2| + |z__1| < 1$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \sum_{k = 0}^{\infty} \frac{{(a)}_{k} {(b__2)}_{k} {}_{2}F_{1} (a + k, b__1; c__1; z__1) {z__2}^{k}}{{(c__2)}_{k} k!}$

$|z__2| + |z__1| < 1$

series

$series (F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2), z__1, 4) = {}_{2}F_{1} (a, b__2; c__2; z__2) + \frac{a b__1 {}_{2}F_{1} (b__2, a + 1; c__2; z__2)}{c__1} z__1 + \frac{1}{2} \frac{a b__1 (a + 1) (b__1 + 1) {}_{2}F_{1} (b__2, a + 2; c__2; z__2)}{c__1 (c__1 + 1)} {z__1}^{2} + \frac{1}{6} \frac{a b__1 (a + 1) (b__1 + 1) (a + 2) (b__1 + 2) {}_{2}F_{1} (b__2, a + 3; c__2; z__2)}{c__1 (c__1 + 1) (c__1 + 2)} {z__1}^{3} + O ({z__1}^{4})$

$series (F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2), z__2, 4) = {}_{2}F_{1} (a, b__1; c__1; z__1) + \frac{a b__2 {}_{2}F_{1} (b__1, a + 1; c__1; z__1)}{c__2} z__2 + \frac{1}{2} \frac{a b__2 (a + 1) (b__2 + 1) {}_{2}F_{1} (b__1, a + 2; c__1; z__1)}{c__2 (c__2 + 1)} {z__2}^{2} + \frac{1}{6} \frac{a b__2 (a + 1) (b__2 + 1) (a + 2) (b__2 + 2) {}_{2}F_{1} (b__1, a + 3; c__1; z__1)}{c__2 (c__2 + 1) (c__2 + 2)} {z__2}^{3} + O ({z__2}^{4})$

integral form

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{Γ (c__1) (\int_{0}^{1} \frac{u^{b__1 - 1} {}_{2}F_{1} (a, b__2; c__2; - \frac{z__2}{u z__1 - 1})}{{(1 - u)}^{- c__1 + b__1 + 1} {(- u z__1 + 1)}^{a}} &DifferentialD; u)}{Γ (b__1) Γ (c__1 - b__1)}$

$z__1 \neq 1 \land 0 < ℜ (b__1) \land 0 < - ℜ (- c__1 + b__1)$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{Γ (c__2) (\int_{0}^{1} \frac{u^{b__2 - 1} {}_{2}F_{1} (a, b__1; c__1; - \frac{z__1}{u z__2 - 1})}{{(1 - u)}^{1 + b__2 - c__2} {(- u z__2 + 1)}^{a}} &DifferentialD; u)}{Γ (b__2) Γ (c__2 - b__2)}$

$z__2 \neq 1 \land 0 < ℜ (b__2) \land 0 < - ℜ (- c__2 + b__2)$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{Γ (c__1) Γ (c__2) (\int_{0}^{1} \int_{0}^{1} \frac{u^{b__1 - 1} v^{b__2 - 1}}{{(1 - u)}^{- c__1 + b__1 + 1} {(1 - v)}^{1 + b__2 - c__2} {(- u z__1 - v z__2 + 1)}^{a}} &DifferentialD; u &DifferentialD; v)}{Γ (b__1) Γ (b__2) Γ (c__1 - b__1) Γ (c__2 - b__2)}$

$0 < ℜ (b__1) \land 0 < ℜ (b__2) \land 0 < - ℜ (- c__1 + b__1) \land 0 < - ℜ (- c__2 + b__2)$

$F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{\int_{0}^{\infty} \frac{u^{a - 1} {}_{1}F_{1} (b__1; c__1; u z__1) {}_{1}F_{1} (b__2; c__2; u z__2)}{{&ExponentialE;}^{u}} &DifferentialD; u}{Γ (a)}$

$ℜ (z__1 + z__2) < 1 \land 0 < ℜ (a)$

differentiation rule

$\frac{\partial}{\partial z__1} F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{a b__1 F_{2} (a + 1, b__1 + 1, b__2, c__1 + 1, c__2, z__1, z__2)}{c__1}$

$\frac{\partial^{n}}{\partial {z__1}^{n}} F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{{(a)}_{n} {(b__1)}_{n} F_{2} (n + a, n + b__1, b__2, n + c__1, c__2, z__1, z__2)}{{(c__1)}_{n}}$

$\frac{\partial}{\partial z__2} F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{a b__2 F_{2} (a + 1, b__1, b__2 + 1, c__1, c__2 + 1, z__1, z__2)}{c__2}$

$\frac{\partial^{n}}{\partial {z__2}^{n}} F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{{(a)}_{n} {(b__2)}_{n} F_{2} (n + a, b__1, n + b__2, c__1, n + c__2, z__1, z__2)}{{(c__2)}_{n}}$

$f (a, b__1, b__2, c__1, c__2, z__1, z__2) = F_{2} (a, b__1, b__2, c__1, c__2, z__1, z__2)$

$\frac{\partial^{2}}{\partial {z__1}^{2}} f (a, b__1, b__2, c__1, c__2, z__1, z__2) = - \frac{z__2 (\frac{\partial^{2}}{\partial z__2 \partial z__1} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1 - 1} + \frac{((- a - b__1 - 1) z__1 + c__1) (\frac{\partial}{\partial z__1} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1 (z__1 - 1)} - \frac{b__1 z__2 (\frac{\partial}{\partial z__2} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1 (z__1 - 1)} - \frac{f (a, b__1, b__2, c__1, c__2, z__1, z__2) a b__1}{z__1 (z__1 - 1)}$

$\frac{\partial^{2}}{\partial z__2 \partial z__1} f (a, b__1, b__2, c__1, c__2, z__1, z__2) = \frac{(1 - z__2) (\frac{\partial^{2}}{\partial {z__2}^{2}} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1} - \frac{b__2 (\frac{\partial}{\partial z__1} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__2} + \frac{((- a - b__2 - 1) z__2 + c__2) (\frac{\partial}{\partial z__2} f (a, b__1, b__2, c__1, c__2, z__1, z__2))}{z__1 z__2} - \frac{f (a, b__1, b__2, c__1, c__2, z__1, z__2) a b__2}{z__1 z__2}$

References

[1] Appell, P.; Kampe de Feriet, J. Fonctions Hypergeometriques et hyperspheriques. Gauthier-Villars, 1926.

[2] Srivastava, H. M.; Karlsson, P. W. Multiple Gaussian Hypergeometric Series. Ellis Horwood, 1985.

Compatibility

•	The AppellF2 command was introduced in Maple 2017.

•	For more information on Maple 2017 changes, see Updates in Maple 2017.

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