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

§16.16 Transformations of Variables

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Keywords:
Appell functions, transformations of variables
Permalink:
http://dlmf.nist.gov/16.16
See also:
Annotations for Ch.16
Contents

§16.16(i) Reduction Formulas

16.16.1 F1⁡(α;β,β′;β+β′;x,y) =(1-y)-α⁢F12⁡(α,ββ+β′;x-y1-y),
16.16.2 F2⁡(α;β,β′;γ,β′;x,y) =(1-y)-α⁢F12⁡(α,βγ;x1-y),
16.16.3 F2⁡(α;β,β′;γ,α;x,y) =(1-y)-β′⁢F1⁡(β;α-β′,β′;γ;x,x1-y),
16.16.4 F3⁡(α,γ-α;β,β′;γ;x,y) =(1-y)-β′⁢F1⁡(α;β,β′;γ;x,yy-1),
16.16.5 F3⁡(α,γ-α;β,γ-β;γ;x,y) =(1-y)α+β-γ⁢F12⁡(α,βγ;x+y-x⁢y),
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′)k⁢k!⁢xk⁢yk⁢F12⁡(α+k,β+kγ+k;x)⁢F12⁡(α+k,β+kγ′+k;y);

see Burchnall and Chaundy (1940, 1941).

§16.16(ii) Other Transformations

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Keywords:
Appell functions, generalized hypergeometric functions, quadratic, transformations of variables, with two variables
Notes:
See Erdélyi et al. (1953a, §5.11).
Permalink:
http://dlmf.nist.gov/16.16.ii
See also:
Annotations for §16.16 and Ch.16
16.16.8 F1⁡(α;β,β′;γ;x,y)=(1-x)-β⁢(1-y)-β′⁢F1⁡(γ-α;β,β′;γ;xx-1,yy-1)=(1-x)-α⁢F1⁡(α;γ-β-β′,β′;γ;xx-1,y-x1-x),
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Symbols:
F1⁡(α;β,β′;γ;x,y): first Appell function
Permalink:
http://dlmf.nist.gov/16.16.E8
Encodings:
TeX, pMML, png
See also:
Annotations for §16.16(ii), §16.16 and Ch.16
16.16.9 F2⁡(α;β,β′;γ,γ′;x,y)=(1-x)-α⁢F2⁡(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)-α⁢F4⁡(α,α-γ′+1;γ,α-β+1;xy,1y)+Γ⁡(γ′)⁢Γ⁡(α-β)Γ⁡(γ′-β)⁢Γ⁡(α)⁢(-y)-β⁢F4⁡(β,β-γ′+1;γ,β-α+1;xy,1y).

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

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