A series transformation formula and related polynomials (Q2368504)

The author establishes an important expression for a power series in terms of a series of functions: \[ \sum_{n=0 }^\infty g^{(n)}(0) f(n) x^n = \sum_{n=0 }^\infty f^{(n)}(0)/n! \sum_{k=0 }^n S(n,k) x^k g^ {(k)} (x) \] where \(f\) and \(g\) are appropriate functions and the \(S(n,k)\) are the Stirling numbers of the second kind. When \(f\) is a polynomial, it evaluates the left hand side in a closed form. This representation transforms certain convergent series into asymptotic series. When \(g(x) = e^x\), Ramanujan [see \textit{B. C. Berndt}, Ramanujan's notebooks. Part I (Springer, New York etc.) (1985; Zbl 0555.10001)] presented several interesting applications.