Some Results On The Bell Matrices And Their Applications

The main purpose of this thesis is to study the Bell matrices，and obtain therelationships between the Bell matrices，the generalized Pascal functional matricesand the generalized Riordan arrays．By substituting the special sequences for thevariables of the Bell polynomials，some combinatorial identities and usefulinverse formulas with respect to some special sequences are also derived．Thecontents can be summarized as follows：Chapter1introduces the backgrounds and the current researches of Bellpolynomials，the generalized Pascal functional matrices and the generalizedRiordan array．Moreover, it introduces the matrix method provides a powerful toolfor the combinatorial identities．The following two chapters are the results of thepresent thesis．Chapter2can be divided into two parts．The first part of Chapter2is devoted to the Bell matrices and thegeneralized Pascal functional matrices by using the matrix method，the generatingfunctions and so on． Moreover， some combinatorial identities are also derivedfrom the corresponding matrix representations．In the second part of Chapter2, the main purpose is to obtain thefactorization of the matrices related to the ordinary Bell polynomials by using thegenerating functions and the unified approaches to the factorizations of manylower triangular matrices．Moreover，some combinatorial identities are alsoderived from the corresponding matrix representations．In chapter3，the main purpose is to study the relationship of the generalizedBell matrices and the generalized Riordan matrices，and obtain the inverseformula of the Bell polynomials with respect to，and then deduce severalinverse formulas using the generating functions and the inverse formula ofLagrange．...