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, ...

We analyze the Bell polynomials Bn(x) asymptotically as n → ∞. We obtain asymptotic approximations from the differential-difference equation which they satisfy, using a discrete version of the ray method. We give some examples showing the accuracy of our formulas.

Letting B n (x) the n-th Bell polynomial, it is well known that B n admit specific integer coordinates in the two following bases x i numbers and binomial coefficients. Our aim is to prove that, for r + s = n, the sequence x j B k (x) is a family of bases of the Q-vectorial space formed by polynomials of Q [X ] for which B n admits a Binomial Recurrence Coefficient.

Chou, Hsu and Shiue gave some applications of Faà di Bruno’s formula for the characterization of inverse relations. In this paper, we use partial Bell polynomials and binomial-type sequence of polynomials to develop complementary inverse relations.