Three explicit formulas for the Taylor coefficients of the function \(({{1-z} \over {1-xz}} )^ \lambda\) (Q1913322)

The author develops some nice formulas for the expansion of the function \[ \Biggl( {1- z\over 1- xz}\Biggr)^\lambda= 1+ \sum^\infty_{n= 1} C_n(\lambda, x) z^n,\quad |z|< 1 \] and also \(|x|= 1\), \(\lambda\) arbitrary complex numbers. These calculations (that are clever!) enable to find some solutions to open questions concerning the extremal of certain classes of functions. In particular -- a conjecture of Silverman and Silvia concerning functions starlike with respect to a boundary point -- is solved for \(n= 4\).