Lagrange-b¨¹rmann Expansion Theorem And The Inversion Formula New Exploration

This present thesis is concerned with the Lagrange–Bu¨rmann expansion theoremsubject to two diferent conditions: explicit and implicit functional equations. Themain methods are divided into analytic and algebraic method (or formal power series).Some typical applications to combinatorial identities and new matrix inversions arepresented. It includes the following four parts:Chapter one is devoted to a brief introduction of Combinatorial inversions and therelationship between Lagrange–Bu¨rmann expansion theorem and inversion Formula.Some necessary terminologies and notations, together with our purpose, are introduced.In Chapter two, we discuss the Lagrange–Bu¨rmann expansion theorem, accordingto one variable to multivariate version from the viewpoint of analysis and algebra. Asone of main results of this thesis, we present two new proofs of one-and multi-variatethe Lagrange–Bu¨rmann expansion theorem. Our methods reveal that the Lagrange–Bu¨rmann expansion theorem boils down to a basic proposition of determinants.Chapter three is devoted to the problems of the Lagrange–Bu¨rmann expansiontheorem subject to the condition of implicit functional equations, from one variable tomultivariate forms. As a main result, we obtain a new pair of matrix inversions.The last chapter is devoted to discrete analogous of the multivariate Lagrange–Bu¨rmann expansion theorem. Inspired by Ma's argument on the (f, g)–inversion for-mula and using the similar argument as the preceding chapter, we obtain a new proof ofthe multivariate matrix inversion, which originally due to Krattenthaler and Schlosser.It is also one of results in the present thesis.