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DLMF: §16.13 Appell Functions ‣ Two-Variable Hypergeometric Functions ‣ Chapter 16 Generalized Hypergeometric Functions and Meijer 𝐺-Function
§16.13 Appell Functions
The following four functions of two real or complex variables x and y
cannot be expressed as a product of two F 1 2 functions, in
general, but they satisfy partial differential equations that resemble the
hypergeometric differential equation (15.10.1 ):
16.13.1
F 1 ( α ; β , β ′ ; γ ; x , y )
= ∑ m , n = 0 ∞ ( α ) m + n ( β ) m ( β ′ ) n ( γ ) m + n m ! n ! x m y n ,
max ( | x | , | y | ) < 1 ,
16.13.2
F 2 ( α ; β , β ′ ; γ , γ ′ ; x , y )
= ∑ m , n = 0 ∞ ( α ) m + n ( β ) m ( β ′ ) n ( γ ) m ( γ ′ ) n m ! n ! x m y n ,
| x | + | y | < 1 ,
16.13.3
F 3 ( α , α ′ ; β , β ′ ; γ ; x , y )
= ∑ m , n = 0 ∞ ( α ) m ( α ′ ) n ( β ) m ( β ′ ) n ( γ ) m + n m ! n ! x m y n ,
max ( | x | , | y | ) < 1 ,
16.13.4
F 4 ( α , β ; γ , γ ′ ; x , y )
= ∑ m , n = 0 ∞ ( α ) m + n ( β ) m + n ( γ ) m ( γ ′ ) n m ! n ! x m y n ,
| x | + | y | < 1 .
Here and elsewhere it is assumed that neither of the bottom parameters
γ and γ ′ is a nonpositive integer.
For large parameter asymptotics see López et al. (2013a , b ) ,
and Ferreira et al. (2013a , b ) .