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.
Overview, HypExplicit, HypergeometricIR, HypergeometricSE, ToHypergeometric.
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)