A kind of asymptotic expansion using partitions (Q1177409)

Let \(\varphi(t)=\sum_{n\geq 0}a_ nt^ n\) be a formal power series with \(a_ 0=\varphi(0)=1\). For any \(\alpha\in C\) we have \((\varphi(t))^ \alpha=\sum_{n\geq 0}{\alpha\brace n}t^ n\), where \[ {\alpha\brace n}={1\over n!}D^ n_ t(\varphi(t))^ \alpha\mid_{t=1},\quad D=d/dt. \] A certain algebraic identity involving a summation over partitions is used to determine an asymptotic expansion of the function \({\lambda\alpha\brace n}\) in terms of \(\lambda\in C\) as \(|\lambda|\to\infty\). Some applications related to well-known number sequences and polynomials are presented.