Applications Of Inverse Relations In WZ Theory And Combinatorica Identities

In this thesis, we mainly discuss the applications of matrix representation of inverse relations to the WZ theory and to the hypergeometric series identities.Chapter one is concerned with the basic knowledge of inverse relations, its applications in the Combinatorics. Moreover introduction to the Gould-Hsu inversion, the Gould-Hsu-Carlitz inversion and the Krattenthaler inversion.Chapter two is concerned with the interpolation-typed matrix inversion that is developed by professor Ma. We prove two interpolation-typed matrix inversions as (f,f) inversion and (f, g) inversion which including all the matrix inversions in chapter one by induction.As the main part of this paper,Chapter three describes Milne's characterization theorem, which can be restated as the matrix equation AX=XB and the WZ method from the matrix operation. We derive new WZ pairs from the (f, g) inversion and a known one.In the last chapter we derive new identities from (f,g)inversion and famous hypergeometric series identities.