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

§16.16 Transformations of Variables

Contents

§16.16(i) Reduction Formulas

16.16.1 F1(α;β,β;β+β;x,y) =(1-y)-αF12(α,ββ+β;x-y1-y),
16.16.2 F2(α;β,β;γ,β;x,y) =(1-y)-αF12(α,βγ;x1-y),
16.16.3 F2(α;β,β;γ,α;x,y) =(1-y)-βF1(β;α-β,β;γ;x,x1-y),
16.16.4 F3(α,γ-α;β,β;γ;x,y) =(1-y)-βF1(α;β,β;γ;x,yy-1),
16.16.5 F3(α,γ-α;β,γ-β;γ;x,y) =(1-y)α+β-γF12(α,βγ;x+y-xy),
16.16.6 F4(α;β;γ,α+β-γ+1;x(1-y),y(1-x)) =F12(α,βγ;x)F12(α,βα+β-γ+1;y).

See Erdélyi et al. (1953a, §5.10) for these and further reduction formulas. An extension of (16.16.6) is given by

16.16.7 F4(α;β;γ,γ;x(1-y),y(1-x))=k=0(α)k(β)k(α+β-γ-γ+1)k(γ)k(γ)kk!xkykF12(α+k,β+kγ+k;x)F12(α+k,β+kγ+k;y);

see Burchnall and Chaundy (1940, 1941).

§16.16(ii) Other Transformations

16.16.8 F1(α;β,β;γ;x,y)=(1-x)-β(1-y)-βF1(γ-α;β,β;γ;xx-1,yy-1)=(1-x)-αF1(α;γ-β-β,β;γ;xx-1,y-x1-x),
16.16.9 F2(α;β,β;γ,γ;x,y)=(1-x)-αF2(α;γ-β,β;γ,γ;xx-1,y1-x),
16.16.10 F4(α;β;γ,γ;x,y)=Γ(γ)Γ(β-α)Γ(γ-α)Γ(β)(-y)-αF4(α;α-γ+1;γ,α-β+1;xy,1y)+Γ(γ)Γ(α-β)Γ(γ-β)Γ(α)(-y)-βF4(β;β-γ+1;γ,β-α+1;xy,1y).

For quadratic transformations of Appell functions see Carlson (1976).