The Algebraic Structure Of The (f,g)-inversion

This thesis is mainly concerned with the algebraic structure of the (f, g)-inversion.In Chapter one, we make a brief introduction about the definition of inversion relation, some classical inversion relations, as well as the purpose of studying inversion relations. Among these are the Gould-Hsu inversion, the Krattenthaler inversion, the Gould-Hsu-Carlitz inversion formula, the Bailey lemma, Bressoud's inversion.In Chapter two, we introduce the concept of the interpolation inversion and use it to set up the (f, g)-inversion. We also present another proof of the (f,f)-inversion by using Krattenthaler's operator method.Chapter three is devoted to the problem of finding the explicit expressions of f and g such thatg{a, b)f(x, c) - g{a, c)f(x, b) + g(b, c)f(x, a) = 0.with the assumption that f and g are polynomials or infinite series. As we will see later, such a pair of functions f and g always leads us to the (f, g) inversion due to Ma (cf.[9]), which is of value to the basic hypergeometric series.In Chapter four, we make further investigation on the so-called inheritable property of the set of solutions of the above equation. Furthermore, we find there exists a bijective relation between this set and the set of antisymmetric matrices in the case f = g. This bijection makes the (f,g)- in version conveniently to use.As applications of main theorems in Chapter three, the last chapter provides some reproofs about some remarkable inversions in Combinatorics, among these are Gasper's bibasic inversion and Chu's bivariate form of Gould-Hsu inversion. It is worth noting that our proofs are simpler and shorter than ones previously known.