HypergeometricIR[exp, t]
substitutes for all Hypergeometric2F1[a,b,c,z]
in exp
by its Euler integral representation. The factor Integratedx[t, 0, 1]
can be omitted by setting the option Integratedx -> False
.
Overview, HypergeometricAC, HypergeometricSE, ToHypergeometric.
[Hypergeometric2F1[a, b, c, z], t] HypergeometricIR
[t^b (1 - t)^c (1 + t z)^a, t]
ToHypergeometric
[%, t]
HypergeometricIR
\frac{\Gamma (b+1) \Gamma (c+1) \, _2F_1(-a,b+1;b+c+2;-z)}{\Gamma (b+c+2)}