Asymptotic expansion of some Sheffer polynomials (Q2774165)

Sheffer polynomials are generated by functions of the form NEWLINE\[NEWLINEA(t)e^{zg(t)}= \sum^\infty_{n=0} P_n(t)t^n,NEWLINE\]NEWLINE where \(A(t)\) and \(g(t)\) are analytic functions on some domain containing zero with \(A(0)=1\), \(g(0)=0\) and \(g' (0)\neq 0\). The importance of these polynomials lies in their being the coefficients of power series expansion of analytic functions. In this paper the \textit{L. C. Hsu}'s method [Approximation Theory Appl. 11, No. 1, 94-104 (1995; Zbl 0827.41024)] (or the cycle-indicator method) is written as a formal theorem and successfully applied to the following Sheffer polynomials: Poisson-Charlier, weighted Touchard, Toscano, and Angelescu polynomials, for to obtain an asymptotic expansion as \(\lambda\) is a parameter tending to positive infinity under some restrictions with respect to the degree of the given polynomials.