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

§16.16 Transformations of Variables

Contents
  1. §16.16(i) Reduction Formulas
  2. §16.16(ii) Other Transformations

§16.16(i) Reduction Formulas

16.16.1 F1(α;β,β;β+β;x,y) =(1y)αF12(α,ββ+β;xy1y),
16.16.2 F2(α;β,β;γ,β;x,y) =(1y)αF12(α,βγ;x1y),
16.16.3 F2(α;β,β;γ,α;x,y) =(1y)βF1(β;αβ,β;γ;x,x1y),
16.16.4 F3(α,γα;β,β;γ;x,y) =(1y)βF1(α;β,β;γ;x,yy1),
16.16.5 F3(α,γα;β,γβ;γ;x,y) =(1y)α+βγF12(α,βγ;x+yxy),
16.16.5_5 F4(α,β;γ,β;x(1y),y(1x)) =(1x)α(1y)αF1(α;γβ,αγ+1;γ;xx1,xy(1x)(1y)),
16.16.6 F4(α,β;γ,α+βγ+1;x(1y),y(1x)) =F12(α,βγ;x)F12(α,βα+βγ+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(1y),y(1x))=k=0(α)k(β)k(α+βγγ+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)=(1x)β(1y)βF1(γα;β,β;γ;xx1,yy1)=(1x)αF1(α;γββ,β;γ;xx1,yx1x),
16.16.9 F2(α;β,β;γ,γ;x,y)=(1x)αF2(α;γβ,β;γ,γ;xx1,y1x),
16.16.10 F4(α,β;γ,γ;x,y)=Γ(γ)Γ(βα)Γ(γα)Γ(β)(y)αF4(α,αγ+1;γ,αβ+1;xy,1y)+Γ(γ)Γ(αβ)Γ(γβ)Γ(α)(y)βF4(β,βγ+1;γ,βα+1;xy,1y).

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