FeynCalc manual (development version)

HypergeometricAC

HypergeometricAC[n][exp] analytically continues Hypergeometric2F1 functions in exp. The second argument n refers to the equation number (n) in chapter 2.10 of “Higher Transcendental Functions” by Erdelyi, Magnus, Oberhettinger, Tricomi. In case of eq. (6) (p.109) the last line is returned for HypergeometricAC[6][exp], while the first equality is given by HypergeometricAC[61][exp].

(2.10.1) is identical to eq. (9.5.7) of “Special Functions & their Applications” by N.N.Lebedev.

See also

Overview, HypExplicit, HypergeometricIR, HypergeometricSE, ToHypergeometric.

Examples

These are all transformation rules currently built in.

HypergeometricAC[1][Hypergeometric2F1[\[Alpha], \[Beta], \[Gamma], z]]

\frac{\Gamma (\gamma ) \Gamma (\alpha +\beta -\gamma ) (1-z)^{-\alpha -\beta +\gamma } \, _2F_1(\gamma -\alpha ,\gamma -\beta ;-\alpha -\beta +\gamma +1;1-z)}{\Gamma (\alpha ) \Gamma (\beta )}+\frac{\Gamma (\gamma ) \Gamma (-\alpha -\beta +\gamma ) \, _2F_1(\alpha ,\beta ;\alpha +\beta -\gamma +1;1-z)}{\Gamma (\gamma -\alpha ) \Gamma (\gamma -\beta )}

HypergeometricAC[2][Hypergeometric2F1[\[Alpha], \[Beta], \[Gamma], z]]

\frac{\Gamma (\gamma ) (-z)^{-\alpha } \Gamma (\beta -\alpha ) \, _2F_1\left(\alpha ,\alpha -\gamma +1;\alpha -\beta +1;\frac{1}{z}\right)}{\Gamma (\beta ) \Gamma (\gamma -\alpha )}+\frac{\Gamma (\gamma ) (-z)^{-\beta } \Gamma (\alpha -\beta ) \, _2F_1\left(\beta ,\beta -\gamma +1;-\alpha +\beta +1;\frac{1}{z}\right)}{\Gamma (\alpha ) \Gamma (\gamma -\beta )}

HypergeometricAC[3][Hypergeometric2F1[\[Alpha], \[Beta], \[Gamma], z]]

\frac{\Gamma (\gamma ) (1-z)^{-\alpha } \Gamma (\beta -\alpha ) \, _2F_1\left(\alpha ,\gamma -\beta ;\alpha -\beta +1;\frac{1}{1-z}\right)}{\Gamma (\beta ) \Gamma (\gamma -\alpha )}+\frac{\Gamma (\gamma ) (1-z)^{-\beta } \Gamma (\alpha -\beta ) \, _2F_1\left(\beta ,\gamma -\alpha ;-\alpha +\beta +1;\frac{1}{1-z}\right)}{\Gamma (\alpha ) \Gamma (\gamma -\beta )}

HypergeometricAC[4][Hypergeometric2F1[\[Alpha], \[Beta], \[Gamma], z]]

\frac{\Gamma (\gamma ) z^{-\alpha } \Gamma (-\alpha -\beta +\gamma ) \, _2F_1\left(\alpha ,\alpha -\gamma +1;\alpha +\beta -\gamma +1;-\frac{1-z}{z}\right)}{\Gamma (\gamma -\alpha ) \Gamma (\gamma -\beta )}+\frac{\Gamma (\gamma ) z^{\alpha -\gamma } \Gamma (\alpha +\beta -\gamma ) (1-z)^{-\alpha -\beta +\gamma } \, _2F_1\left(1-\alpha ,\gamma -\alpha ;-\alpha -\beta +\gamma +1;-\frac{1-z}{z}\right)}{\Gamma (\alpha ) \Gamma (\beta )}

HypergeometricAC[6][Hypergeometric2F1[\[Alpha], \[Beta], \[Gamma], z]]

(1-z)^{-\beta } \, _2F_1\left(\beta ,\gamma -\alpha ;\gamma ;-\frac{z}{1-z}\right)

HypergeometricAC[61][Hypergeometric2F1[\[Alpha], \[Beta], \[Gamma], z]]

(1-z)^{-\alpha } \, _2F_1\left(\alpha ,\gamma -\beta ;\gamma ;-\frac{z}{1-z}\right)