[go: up one dir, main page]
16 Generalized Hypergeometric Functions & Meijer G-FunctionTwo-Variable Hypergeometric Functions

§16.16 Transformations of Variables

ⓘ
Keywords:
Appell functions, transformations of variables
Permalink:
http://dlmf.nist.gov/16.16
See also:
Annotations for Ch.16
Contents
  1. §16.16(i) Reduction Formulas
  2. §16.16(ii) Other Transformations

§16.16(i) Reduction Formulas

ⓘ
Keywords:
Appell functions, reduction formulas, relations to hypergeometric functions, transformations of variables
Permalink:
http://dlmf.nist.gov/16.16.i
Addition (effective with 1.1.12):
Equation (16.16.5_5) was added.
See also:
Annotations for §16.16 and Ch.16
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),
16.16.5_5 F4⁡(α,β;γ,β;x⁢(1−y),y⁢(1−x)) =(1−x)−α⁢(1−y)−α⁢F1⁡(α;γ−β,α−γ+1;γ;xx−1,x⁢y(1−x)⁢(1−y)),
ⓘ
Symbols:
F1⁡(α;β,β′;γ;x,y): first Appell function and F4⁡(α,β;γ,γ′;x,y): fourth Appell function
Referenced by:
§16.16(i), Erratum (V1.1.12) for Additions
Permalink:
http://dlmf.nist.gov/16.16.E5_5
Encodings:
TeX, pMML, png
Addition (effective with 1.1.12):
This equation was added.
See also:
Annotations for §16.16(i), §16.16 and Ch.16
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

ⓘ
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),
ⓘ
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).

© 2010–2026 NIST / Disclaimer / Feedback; Version 1.2.8; Release date 2026-09-15.

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov