Study Of Some Problems On The Multivariable Lagrange-Good Expansion Theorem

This paper is mainly concerned with the multivariable Lagrange-Good expansion theorem.In Chapter two, we mainly analyze all different proofs of the classical Lagrange expansion theorem from the viewpoints of both complex analysis and algebraic combi-nation. All these leads us to a new (at least previously unknown) proof of two transfer formulas in Umbral Calculus and of the multivariable Lagrange-Good expansion the-orem in terms of formal power series.In Chapter three, by means of the multivariable Lagrange-Good expansion theo-rem, we set up a new multidimensional matrix inversion which is one of main results as follows.Theorem(m-multidimensional matrix inversion) Let h(X),φ(X),ψ(X) be ar-bitrary formal power series such that φ(O)ψ(O)≠0. Assume further that zi=xi/φi(X),wi=xiψi(X),1≤i≤m. There holdsAt the same time, other applications of the multivariable Lagrange-Good expan-sion theorem to convolution identities such sort are reconsidered. Some new results are given.