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

§16.14 Partial Differential Equations


§16.14(i) Appell Functions

16.14.1 x(1-x)2F1x2+y(1-x)2F1xy+(γ-(α+β+1)x)F1x-βyF1y-αβF1 =0,
y(1-y)2F1y2+x(1-y)2F1xy+(γ-(α+β+1)y)F1y-βxF1x-αβF1 =0,
16.14.2 x(1-x)2F2x2-xy2F2xy+(γ-(α+β+1)x)F2x-βyF2y-αβF2 =0,
y(1-y)2F2y2-xy2F2xy+(γ-(α+β+1)y)F2y-βxF2x-αβF2 =0,
16.14.3 x(1-x)2F3x2+y2F3xy+(γ-(α+β+1)x)F3x-αβF3 =0,
y(1-y)2F3y2+x2F3xy+(γ-(α+β+1)y)F3y-αβF3 =0,
16.14.4 x(1-x)2F4x2-2xy2F4xy-y22F4y2+(γ-(α+β+1)x)F4x-(α+β+1)yF4y-αβF4 =0,
y(1-y)2F4y2-2xy2F4xy-x22F4x2+(γ-(α+β+1)y)F4y-(α+β+1)xF4x-αβF4 =0.

§16.14(ii) Other Functions

In addition to the four Appell functions there are 24 other sums of double series that cannot be expressed as a product of two F12 functions, and which satisfy pairs of linear partial differential equations of the second order. Two examples are provided by

16.14.5 G2(α,α;β,β;x,y) =m,n=0Γ(α+m)Γ(α+n)Γ(β+n-m)Γ(β+m-n)Γ(α)Γ(α)Γ(β)Γ(β)xmynm!n!,
|x|<1, |y|<1,
16.14.6 G3(α,α;x,y) =m,n=0Γ(α+2n-m)Γ(α+2m-n)Γ(α)Γ(α)xmynm!n!,

(The region of convergence |x|+|y|<14 is not quite maximal.) See Erdélyi et al. (1953a, §§5.7.1–5.7.2) for further information.