We present a formula that turns power series into series of functions. This formula serves two purposes: first, it helps to evaluate some power series in a closed form; second, it transforms certain power series into asymptotic series. For example, we find the asymptotic expansions for λ>0 of the incomplete gamma function γ(λ,x) and of the Lerch transcendent Φ(x,s,λ). In one particular case, our formula reduces to a series transformation formula which appears in the works of Ramanujan and is related to the exponential (or Bell) polynomials. Another particular case, based on the geometric series, gives rise to a new class of polynomials called geometric polynomials

We construct a functional calculus, g H> g(A), for functions, g, that are the sum of a Stieltjes function and a nonnegative operator monotone function, and unbounded linear operators, A, whose resolvent set contains (-oo, 0), with [\\r(r + A)' 1 1| | r > 0} bounded. For such functions g, we show that -g(A) generates a bounded holomorphic strongly continuous semigroup of angle 6, whenever -A does.We show that, for any Bernstein function f,-f(A) generates a bounded holomorphic strongly continuous semigroup of angle 7r/2, whenever -A does.We also prove some new results about the Bochner-Phillips functional calculus. We discuss the relationship between fractional powers and our construction.1991 Mathematics subject classification (Amer. Math. Soc): 47 A 60, 47 B 44, 47 D 05.

This article is a short elementary review of the exponential polynomials (also called single-variable Bell polynomials) from the point of view of Analysis. Some new properties are included and several Analysis-related applications are mentioned. At the end of the paper one application is described in details -certain Fourier integrals involving and are evaluated in terms of Stirling numbers.

This is a short introduction to the theory of Stirling numbers of the second kind ( , ) S m k from the point of view of analysis. It is written in the form of a historical survey. We tell the story of their birth in the book of James Stirling (1730) and show how they mature in the works of Johann Grünert (1843). We demonstrate their usefulness in several differentiation formulas. The reader can also see the connection of ( , ) S m k to Bernoulli numbers, to Euler polynomials and to power sums.Mathematics Subject Classification 2010: 11B83; 05A19.

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