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16 Generalized Hypergeometric Functions and Meijer G-FunctionTwo-Variable Hypergeometric Functions

§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 F12 functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1):

16.13.1 F1(α;β,β;γ;x,y) =m,n=0(α)m+n(β)m(β)n(γ)m+nm!n!xmyn,
max(|x|,|y|)<1,
16.13.2 F2(α;β,β;γ,γ;x,y) =m,n=0(α)m+n(β)m(β)n(γ)m(γ)nm!n!xmyn,
|x|+|y|<1,
16.13.3 F3(α,α;β,β;γ;x,y) =m,n=0(α)m(α)n(β)m(β)n(γ)m+nm!n!xmyn,
max(|x|,|y|)<1,
16.13.4 F4(α;β;γ,γ;x,y) =m,n=0(α)m+n(β)m+n(γ)m(γ)nm!n!xmyn,
|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. (2013).