Transformation of differential operator in Casselman-Milícic paper into hypergeometric form

Note by Tom Koornwinder, T.H.Koornwinder@uva.nl, January 4, 2010

See Example 3.7 in W. Casselman & D. Milicic, Asymptotic behavior of matrix coefficients of admissible representations, Duke Math. J. 49 (1982), 869–930.

Simplify[(((((((-1+z))^2)/(4 z))))^(-(m+n)/4))(((((((1+z))^2)/(4 z))))^(-(m-n)/4))((z^2)D[(((((((-1+z))^2)/(4 z))))^((m+n)/4))(((((((1+z))^2)/(4 z))))^((m-n)/4))f[-((((-1+z))^2)/(4 z))],{z,2}]-((2(z^3))/(1-(z^2))) D[(((((((-1+z))^2)/(4 z))))^((m+n)/4))(((((((1+z))^2)/(4 z))))^((m-n)/4))f[-((((-1+z))^2)/(4 z))],z])-(((n^2)+(m^2))((z^2)/(((-1+(z^2)))^2))+n m ((z(1+(z^2)))/(((-1+(z^2)))^2)))f[-((((-1+z))^2)/(4 z))]==-((w(1-w)f''[w]+((m+n)/2+1-(m+2)w)f'[w]-+m(m+2)/4f[w])/.w->-((((-1+z))^2)/(4 z)))]