The Treatise On The Matrix Representation Of Bivariate Recurrence Sequences And Combinatorial Inversion

In this thesis, we mainly discuss the inverse of arbitrary matrices F = (F(n, k)) and G = (G(n,k)) given by bivariate sequences {F(n,k)} and {G(n,k)}, and almost of them turn out to be the special cases of the (f,g)-inversion.Chapter one is concerned with the basic knowledge of inverse relations, and its applications in Combinatorics. More details are given to the (f,g)-inversion and its special cases: the Gould-Hsu inverse,the Krattenthaler inverse, the Gould-Hsu-Carlitz inverse,the Bailey transform,the Bressoud inverse.As the main part of this paper, Chapter two describes Milne's characterization theorem, which can be described by the matrix equation AX = XB. Based on Milner's idea and motivated by the author's own idea, we reconsider the problem of finding the inverse of the infinite,lower-triangular matrices which defined by two bivariate sequences. During this process, three methods including Milne's operate method are put forward and used frequently.In Chapter three, we introduce a sort of shiftable matrix. By using of Mime's characterization theorem, we find its inverse of shiftable matrix entirely. Further, applying the the same technique, we find the third proof of the (f, g)-inverse when f = g. This proof is different from the previously known proof given by the Krattenthaler's operator method.One generalized Mime's characterization theorem is presented in Chapter four. More presecisely, assuming that F = (F(n,k)) and G = (G(n, k)) are not lower-triangular matrices, it turns out that Milne's characterization theorem is still valid. As a result, we obtain two new inverses and further generalize the Bressoud inverse in a more setting.In the last chapter we can see that our method can not solve all possible prob-lems,and we give one example to illuminate in what cases the method would be of no effect, what's more, we give two new problems to be solved.