The n-divided difference of the composite function h ：= f o g of functions f, g at a group of nodes t0,t1,…,tn is shown by the combinations of divided differences of f at the group of nodes g（t0）,g（t1）,…,g（tm...The n-divided difference of the composite function h ：= f o g of functions f, g at a group of nodes t0,t1,…,tn is shown by the combinations of divided differences of f at the group of nodes g（t0）,g（t1）,…,g（tm） and divided differences of g at several partial group of nodes t0,t1,…,tn, where m = 1,2,…,n. Especially, when the given group of nodes are equal to each other completely, it will lead to Faà di Bruno＇s formula of higher derivatives of function h.展开更多
In this paper, we consider the higher divided difference of a composite function f（g（t）） in which g（t） is an s-dimensional vector. By exploiting some properties from mixed partial divided differences and multiva...In this paper, we consider the higher divided difference of a composite function f（g（t）） in which g（t） is an s-dimensional vector. By exploiting some properties from mixed partial divided differences and multivariate Newton interpolation, we generalize the divided difference form of Faà di Bruno＇s formula with a scalar argument. Moreover, a generalized Faà di Bruno＇s formula with a vector argument is derived.展开更多
The purpose of this paper is to establish a formula of higher derivative byFaa di Bruno formula, and apply it to some known results to get some identities involvingcomplete Bell polynomials.
Let X be a complex Banach space with norm · , B be the unit ball in X, Dn be the unit polydisc in Cn. In this paper, we introduce a class of holomorphic mappings Mg on B or Dn. Let f(x) be a normalized locally bi...Let X be a complex Banach space with norm · , B be the unit ball in X, Dn be the unit polydisc in Cn. In this paper, we introduce a class of holomorphic mappings Mg on B or Dn. Let f(x) be a normalized locally biholomorphic mapping on B such that (Df(x))-1f(x) ∈ Mg and f(x) - x has a zero of order k + 1 at x = 0. We obtain coeffcient estimates for f(x). These results unify and generalize many known results.展开更多
Presented in this paper is a convergence theorem for a kind of composite power series expansions whose coefficients can be expressed by using Faa` di Bruno’s formula. A related problem is proposed as a remark, and a ...Presented in this paper is a convergence theorem for a kind of composite power series expansions whose coefficients can be expressed by using Faa` di Bruno’s formula. A related problem is proposed as a remark, and a few examples are given as applications.展开更多
基金This work was supported by the National Science Foundation of China （Grant No.10471128）.
文摘The n-divided difference of the composite function h ：= f o g of functions f, g at a group of nodes t0,t1,…,tn is shown by the combinations of divided differences of f at the group of nodes g（t0）,g（t1）,…,g（tm） and divided differences of g at several partial group of nodes t0,t1,…,tn, where m = 1,2,…,n. Especially, when the given group of nodes are equal to each other completely, it will lead to Faà di Bruno＇s formula of higher derivatives of function h.
基金Acknowledgments. This work was supported by the National Science Foundation of China （Grant Nos. 10471128, 10731060）.
文摘In this paper, we consider the higher divided difference of a composite function f（g（t）） in which g（t） is an s-dimensional vector. By exploiting some properties from mixed partial divided differences and multivariate Newton interpolation, we generalize the divided difference form of Faà di Bruno＇s formula with a scalar argument. Moreover, a generalized Faà di Bruno＇s formula with a vector argument is derived.
基金Supported by the National Natural Science Foundation of China(Grant No.11601543,No.11601216,11701257)Supported by the NSF of Henan Province under Grant(No.172102410069)+1 种基金Supported by the NSF of Education Bureau of Henan Province under Grant(No.16B110009,18A110025)Supported by the Youth Foundation of Luoyang Normal university under Grant(No.2013-QNJJ-001)
文摘The purpose of this paper is to establish a formula of higher derivative byFaa di Bruno formula, and apply it to some known results to get some identities involvingcomplete Bell polynomials.
基金supported by National Natural Science Foundation of China (Grant No. 10571164)Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20050358052)+1 种基金the Jiangxi Provincial Natural Science Foundation of China (Grant No. 2007GZS0177)Specialized Research Fund for the Doctoral Program of Jiangxi Normal University
文摘Let X be a complex Banach space with norm · , B be the unit ball in X, Dn be the unit polydisc in Cn. In this paper, we introduce a class of holomorphic mappings Mg on B or Dn. Let f(x) be a normalized locally biholomorphic mapping on B such that (Df(x))-1f(x) ∈ Mg and f(x) - x has a zero of order k + 1 at x = 0. We obtain coeffcient estimates for f(x). These results unify and generalize many known results.
文摘Presented in this paper is a convergence theorem for a kind of composite power series expansions whose coefficients can be expressed by using Faa` di Bruno’s formula. A related problem is proposed as a remark, and a few examples are given as applications.
