An infinite series which arises in certain applications of the Lagrange-Bürmann formula to exponential functions is investigated. Several very exact estimates for the Laplace transform and higher moments of this function are developed.

Zeta-generalized-Euler-constant functions, $$\gamma \left(s\right):=\sum _{k=1}^{\infty }\left(\frac{1}{k^s}-{\int }_k^{k+1}\frac{dx}{x^s}\right)$$ and $$\tilde{\gamma }\left(s\right):=\sum _{k=1}^{\infty }{\left(-1\right)}^{k+1}\left(\frac{1}{k^s}-{\int }_k^{k+1}\frac{dx}{x^s}\right)$$ defined on the closed interval [0, ∞), where γ(1) is the Euler-Mascheroni constant and $$\tilde{\gamma }$$ (1) = ln $$\frac{4}{\pi }$$ , are studied and estimated with high accuracy.

An asymptotic approximation of Wallis’ sequence W(n) = Πk=1n 4k 2/(4k 2 − 1) obtained on the base of Stirling’s factorial formula is presented. As a consequence, several accurate new estimates of Wallis’ ratios w(n) = Πk=1n(2k−1)/(2k) are given. Also, an asymptotic approximation of π in terms of Wallis’ sequence W(n) is obtained, together with several double inequalities such as, for example, $W\left(n\right)·(a_n+b_n)<\pi <W\left(n\right)·(a_n+b_n^{\text{'}})$ with $a_n=2+\frac{1}{2n+1}+\frac{2}{3{(2n+1)}^2}-\frac{1}{3n{(2n+1)}^{\text{'}}}{b}_n=\frac{2}{33{(n+1)}^{2^{\text{'}}}}{b}_n^{\text{'}}\frac{1}{13n^{2^{\text{'}}}}n\in \mathbb{N}$ .