Several Research Problems in Analytic Combinatorics 

Author  FangZuo 
Tutor  WangTianMing 
School  Dalian University of Technology 
Course  Applied Mathematics 
Keywords  Selfinverse sequences Umbral calculus Sequences of binomial type Sheffer sets Laguerre polynomials Bernoulli numbers Genocchi numbers Stirling numbers Cauchy numbers Riordan arrays Divided differences Reciprocal series Newton series Symbolic operator Newton generating functions 
CLC  O157.1 
Type  PhD thesis 
Year  2008 
Downloads  105 
Quotes  1 
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 selfinverse sequences related to sequences of polynomials of binomial type and the selfinverse 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.