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

Β§16.14 Partial Differential Equations

Contents
  1. Β§16.14(i) Appell Functions
  2. Β§16.14(ii) Other Functions

Β§16.14(i) Appell Functions

16.14.1 x⁒(1βˆ’x)β’βˆ‚2F1βˆ‚x2+y⁒(1βˆ’x)β’βˆ‚2F1βˆ‚xβ’βˆ‚y+(Ξ³βˆ’(Ξ±+Ξ²+1)⁒x)β’βˆ‚F1βˆ‚xβˆ’Ξ²β’yβ’βˆ‚F1βˆ‚yβˆ’Ξ±β’Ξ²β’F1 =0,
y⁒(1βˆ’y)β’βˆ‚2F1βˆ‚y2+x⁒(1βˆ’y)β’βˆ‚2F1βˆ‚xβ’βˆ‚y+(Ξ³βˆ’(Ξ±+Ξ²β€²+1)⁒y)β’βˆ‚F1βˆ‚yβˆ’Ξ²β€²β’xβ’βˆ‚F1βˆ‚xβˆ’Ξ±β’Ξ²β€²β’F1 =0,
16.14.2 x⁒(1βˆ’x)β’βˆ‚2F2βˆ‚x2βˆ’x⁒yβ’βˆ‚2F2βˆ‚xβ’βˆ‚y+(Ξ³βˆ’(Ξ±+Ξ²+1)⁒x)β’βˆ‚F2βˆ‚xβˆ’Ξ²β’yβ’βˆ‚F2βˆ‚yβˆ’Ξ±β’Ξ²β’F2 =0,
y⁒(1βˆ’y)β’βˆ‚2F2βˆ‚y2βˆ’x⁒yβ’βˆ‚2F2βˆ‚xβ’βˆ‚y+(Ξ³β€²βˆ’(Ξ±+Ξ²β€²+1)⁒y)β’βˆ‚F2βˆ‚yβˆ’Ξ²β€²β’xβ’βˆ‚F2βˆ‚xβˆ’Ξ±β’Ξ²β€²β’F2 =0,
16.14.3 x⁒(1βˆ’x)β’βˆ‚2F3βˆ‚x2+yβ’βˆ‚2F3βˆ‚xβ’βˆ‚y+(Ξ³βˆ’(Ξ±+Ξ²+1)⁒x)β’βˆ‚F3βˆ‚xβˆ’Ξ±β’Ξ²β’F3 =0,
y⁒(1βˆ’y)β’βˆ‚2F3βˆ‚y2+xβ’βˆ‚2F3βˆ‚xβ’βˆ‚y+(Ξ³βˆ’(Ξ±β€²+Ξ²β€²+1)⁒y)β’βˆ‚F3βˆ‚yβˆ’Ξ±β€²β’Ξ²β€²β’F3 =0,
16.14.4 x⁒(1βˆ’x)β’βˆ‚2F4βˆ‚x2βˆ’2⁒x⁒yβ’βˆ‚2F4βˆ‚xβ’βˆ‚yβˆ’y2β’βˆ‚2F4βˆ‚y2+(Ξ³βˆ’(Ξ±+Ξ²+1)⁒x)β’βˆ‚F4βˆ‚xβˆ’(Ξ±+Ξ²+1)⁒yβ’βˆ‚F4βˆ‚yβˆ’Ξ±β’Ξ²β’F4 =0,
y⁒(1βˆ’y)β’βˆ‚2F4βˆ‚y2βˆ’2⁒x⁒yβ’βˆ‚2F4βˆ‚xβ’βˆ‚yβˆ’x2β’βˆ‚2F4βˆ‚x2+(Ξ³β€²βˆ’(Ξ±+Ξ²+1)⁒y)β’βˆ‚F4βˆ‚yβˆ’(Ξ±+Ξ²+1)⁒xβ’βˆ‚F4βˆ‚xβˆ’Ξ±β’Ξ²β’F4 =0.

Β§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⁑(Ξ±,Ξ±β€²;Ξ²,Ξ²β€²;x,y) =βˆ‘m,n=0βˆžΞ“β‘(Ξ±+m)⁒Γ⁑(Ξ±β€²+n)⁒Γ⁑(Ξ²+nβˆ’m)⁒Γ⁑(Ξ²β€²+mβˆ’n)Γ⁑(Ξ±)⁒Γ⁑(Ξ±β€²)⁒Γ⁑(Ξ²)⁒Γ⁑(Ξ²β€²)⁒xm⁒ynm!⁒n!,
|x|<1, |y|<1,
16.14.6 G3⁑(Ξ±,Ξ±β€²;x,y) =βˆ‘m,n=0βˆžΞ“β‘(Ξ±+2⁒nβˆ’m)⁒Γ⁑(Ξ±β€²+2⁒mβˆ’n)Γ⁑(Ξ±)⁒Γ⁑(Ξ±β€²)⁒xm⁒ynm!⁒n!,
|x|+|y|<14.

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