Result 1 can be found in N. G. de Bruijn, *Asymptotic Methods in Analysis*, Dover, New York, 1981, p. 71.

Result 2 follows from a formula of E. M. Wright, though he didn't state it in this form. A discussion of this formula, with a generalization, can be found in  Kassie Archer, Ira M. Gessel, Christina Graves, and Xuming Liang, *Counting acyclic and strong digraphs by descents*, Discrete Math. 343 (2020), 112041, 14 pp. https://doi.org/10.1016/j.disc.2020.112041. See Proposition 8. The arXiv version is 
https://doi.org/10.48550/arXiv.1909.01550.

As Sam and Richard noted, Result 3 is well known. It is equivalent to Euler's definition of the Eulerian polynomials. The combinatorial interpretation of the Eulerian polynomials is a special case of a much more general result of MacMahon.