Linear combinations of probabilities (Q1897866)

Let \(f(z)= \sum^\infty_{k= 0} \varphi_k z^k\) converge in the unit disk \(|z|< 1\). The author considers the characteristics \[ \sigma_m= \Biggl({d^m\over dz^m}\log f(z)\Biggr)_{z= 0},\quad \varphi_{n, k}= {1\over 2\pi i} \int_{|z|= 1/2} f^n(z) z^{- k- 1} dz \] and establishes a formula expressing \(\varphi_{n, k}\) in terms of \(\sigma_m\). Then some arguments are given that when one considers the sums of \(n\) i.i.d. random variables, it is sometimes more convenient to deal with the described characteristics rather than usual semi-invariants or factorial semi-invariants.