Treatise On The Lagrange Expansion Theorem And Combinatorial Inversions

In this thesis,we proceed to set up a universal theory on combinatorial inversions associated with the classical Lagrange expansion theorem.Chapter one is a short survey on the history of development of the Lagrange expansion theorem and combinatorial inversions.At the same time,some preliminary concepts about the ring of formal power series and basic hypergeometric series which will be used later are described.Chapter two is devoted to a comprehensive survey on various forms of the La-grange expansion theorem together corresponding proofs.To that end,we classify these conclusions in light of one variable to multivariate,analysis to algebra.In oth-er words,all Lagrange expansion theorems of one variable to multivariate forms and theirs proofs in analysis and the ring of formal power series are discussed in details.We also mention an important kind of generalization,so called q–analogues.In Chapter three,with believing in that combinatorial inversions belongs to a part of general mathematical inverse problems,we study combinatorial inversions from point of view:linear and nonlinear.At first,based on the Lagrange expansion theorem,we sketch a theoretic framework containing the Riordan group as special case.As direct consequences,we derive two linear inversions—(h;?,?)–matrix inversions from the Lagrange expansion theorem.These two inversions include Chu's inversion in[18]and all the inversions in Riordan[83,Chapters II,III].As a general linear inversion,we summarize all results concerning the(f,g)–inversion in[67,68].As a result,we find a new solution for(?)which is sufficient and necessary to the(f,g)–inversion.We also prove a multidimen-sional matrix inversion in[60]due to Krattenthaler and Schlosser via use of difference method.As for nonlinear inversions,we study three kinds of typical cases:a typical inver-sion derived from the Fa(?) di Bruno formula;the inversions among different bases of symmetric polynomial space;and the coefficients of the following sum–product identity(?)In Chapter four,combining the generalized Lagrange inversion in[69]and the Lagrange inversion in[27,43]due to Garsia et al,we give a natural and clear combi-national interpretation between the theory of Dyck path and the Lagrange expansion theorem.As main results,we introduce the so called weighted Dyck path and establish a general Lagrange expansion theorem for expanding any formal power series in terms of any given weighted Dyck path.In Chapter five,we focus on some applications of the Lagrange expansion theorem to establishment of combinatorial inversions and identities.As main results,a unified proof not only for some classical derivative identities in[54]explored by many scholars like Pfaff,Cauchy,Olver et al.but also for their multivariate versions are presented.Some nonlinear inversions associated with a problem proposed by Tao are obtained.We also discuss the validity of the Lagrange expansion theorem under an(unusual)condition(with respect to x=tR(x)),i.e.subject to x=R(x).Our methods and conclusions therein may be regarded as further study on the problems proposed in the paper[35]by Ira M.Gessel in 2016.Finally,we investigate some possibly generalizations for Catalan numbers by use of the Lagrange expansion theorem.