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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
Quote: 1
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Abstract
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.

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