A Study On Some Problems Concerning Combinatorial Inversions

In this thesis, we are mainly concerned with applications of Milne's characteriza-tion theorem stated in matrix form to the theory of combinatorial inversions.Chapter one is a general introduction of short history, some important resultsscatted in many fields of combinatorics and special functions about combinatorial in-versions. Also, the (f, g)-inversion and Milne's characterization theorem are describedin more details since they are main topics in the next two chapters.As the main part of this thesis, Chapter two sets up a few new properties of twofunctions f and g which form an (f, g)-inversion. More interesting is by using thesebasic properties, we successfully found elementary proofs of the summation identitiesof the sine function and elliptic function which were put forward by Chu Wenchangin 1992. Finally, two complete proofs of the (f, g)-inversion by matrix operator andKrattenthaler's operator method are given.Combining Milne's characterization theorem on combinatorial inverse relation withall results obtained in Chapter two, in Chapter three, we further investigate the (α,β)-inversion originally due to Ma xinrong and Xulizhi in 2005 and show by an coun-terexample that the condition in the (α,β)-inversion is only sufficient. By the similarargument, we reprove the classical Lagrange inversion and a result of Roger on Riordanarray.