Several Research Problems in Analytic Combinatorics

In this thesis, we study some aspects in analytic combinatorics. The main contents can be summarized as follows.In Chapter 2, by classical umbral method, we obtain some properties and identities of invariant sequences. More generally, we study the self-inverse sequences related to sequences of polynomials of binomial type and the self-inverse sequences related to Sheffer sets, and give some interesting results of these sequences.In Chapter 3, using generating functions and Riordan arrays, we establish some identities involving Genocchi numbers、Stirling numbers and Cauchy numbers, and give the asymptotics of some combinatorial sums.In Chapter 4, using Newton series, we obtain a divided differences formula of composite functions, and give a matrix equivalent form of these composite functions. Moreover, we derive a divided differences for the reciprocal series.We present several symbolic operator expansions in the last chapter. Furthermore, we give some series transform formulas and Newton generating functions.