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A series transformation formula and related polynomials. (English) Zbl 1086.05006
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 B. C. Berndt, Ramanujan’s notebooks. Part I (Springer, New York etc.) (1985; Zbl 0555.10001)] presented several interesting applications.

05A15 Exact enumeration problems, generating functions
11B73 Bell and Stirling numbers
33C90 Applications of hypergeometric functions
40A30 Convergence and divergence of series and sequences of functions
Zbl 0555.10001
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