Applied Mathematics and Computation, ISSN 0096-3003, 03/2011, Volume 217, Issue 13, pp. 6286 - 6295

A new algorithm for computing the multivariate Faà di Bruno's formula is provided. We use a symbolic approach based on the classical umbral calculus that turns...

Faà di Bruno's formula | Multivariate Hermite polynomial | Multivariate cumulant | Multivariate composite function | Classical umbral calculus | POLYNOMIALS | MATHEMATICS, APPLIED | DIBRUNO,FAA | Faa di Bruno's formula | CUMULANTS | Mathematical models | Calculus | Algorithms | Computation | Mathematical analysis

Faà di Bruno's formula | Multivariate Hermite polynomial | Multivariate cumulant | Multivariate composite function | Classical umbral calculus | POLYNOMIALS | MATHEMATICS, APPLIED | DIBRUNO,FAA | Faa di Bruno's formula | CUMULANTS | Mathematical models | Calculus | Algorithms | Computation | Mathematical analysis

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 03/2010, Volume 59, Issue 6, pp. 2047 - 2052

The well-known formula of Faà di Bruno's for higher derivatives of a composite function has played an important role in combinatorics. In this paper we...

Divided difference | Bell polynomial | Faà di Bruno's formula | Multicomposite function | MATHEMATICS, APPLIED | DIBRUNO,FAA FORMULA | Faa di Bruno's formula | DERIVATIVES

Divided difference | Bell polynomial | Faà di Bruno's formula | Multicomposite function | MATHEMATICS, APPLIED | DIBRUNO,FAA FORMULA | Faa di Bruno's formula | DERIVATIVES

Journal Article

2017, Synthesis lectures on visual computing: computer graphics, animation, computational photography, and imaging, ISBN 9781627059053, Volume 25, xv, 233 pages

This book is written for students, CAD system users and software developers who are interested in geometric continuity-a notion needed in everyday practice of...

Geometry, Differential | Computing and Processing | General Topics for Engineers

Geometry, Differential | Computing and Processing | General Topics for Engineers

Book

Discrete Mathematics, ISSN 0012-365X, 03/2011, Volume 311, Issue 6, pp. 387 - 392

Fa di Bruno's formula is the higher chain rule for differentiation. By means of Gessel's q-composition we derive a q-analogue of Fa di Bruno's determinant...

q-analogue | Complete Bell polynomial | Fa di Bruno's formula | Determinant | MATHEMATICS | DIVIDED DIFFERENCE FORM | Faa di Bruno's formula | Bells | Composite functions | Mathematical analysis | Chains | Determinants | Derivatives | Differentiation

q-analogue | Complete Bell polynomial | Fa di Bruno's formula | Determinant | MATHEMATICS | DIVIDED DIFFERENCE FORM | Faa di Bruno's formula | Bells | Composite functions | Mathematical analysis | Chains | Determinants | Derivatives | Differentiation

Journal Article

Applied Mathematics and Computation, ISSN 0096-3003, 05/2015, Volume 258, pp. 597 - 607

In the paper, the authors first inductively establish explicit formulas for derivatives of the arc sine function, then derive from these explicit formulas...

Derivative | Bell polynomial | Elementary function | Explicit formula | Faá di Bruno formula | MATHEMATICS, APPLIED | Faa di Bruno formula | STIRLING NUMBERS | BERNOULLI | 2ND KIND | FORMULAS

Derivative | Bell polynomial | Elementary function | Explicit formula | Faá di Bruno formula | MATHEMATICS, APPLIED | Faa di Bruno formula | STIRLING NUMBERS | BERNOULLI | 2ND KIND | FORMULAS

Journal Article

Mathematics of Computation of the American Mathematical Society, ISSN 0025-5718, 04/2007, Volume 76, Issue 258, pp. 867 - 877

In this paper we derive two formulas for divided differences of a function of a function. Both formulas lead to other divided difference formulas, such as...

Integers | Interpolation | Composite functions | Product rule | Quotient rule | Chain rule | Polynomials | Mathematical functions | Coefficients | Divided differences | Faà di Bruno's formula | INTERPOLATION | MATHEMATICS, APPLIED | chain rule | Faa di Bruno's formula | divided differences

Integers | Interpolation | Composite functions | Product rule | Quotient rule | Chain rule | Polynomials | Mathematical functions | Coefficients | Divided differences | Faà di Bruno's formula | INTERPOLATION | MATHEMATICS, APPLIED | chain rule | Faa di Bruno's formula | divided differences

Journal Article

Turkish Journal of Mathematics, ISSN 1300-0098, 2014, Volume 38, Issue 3, pp. 558 - 575

We give moment equalities for sums of independent and identically distributed random variables including, in particular, centered and specifically symmetric...

Bootstrap | Faà di bruno's chain rule | Integer partitions | Marcinkiewicz-Zygmund inequalities | Moments | Self-normalized sums | MATHEMATICS | Faa. di Bruno's chain rule | bootstrap | integer partitions | self-normalized sums

Bootstrap | Faà di bruno's chain rule | Integer partitions | Marcinkiewicz-Zygmund inequalities | Moments | Self-normalized sums | MATHEMATICS | Faa. di Bruno's chain rule | bootstrap | integer partitions | self-normalized sums

Journal Article

Discrete Applied Mathematics, ISSN 0166-218X, 01/2020, Volume 272, pp. 90 - 99

In the paper we introduce and study partitions of vectors in Np, as a natural extension of the classical partitions of integers. We show that these vector...

Multi-dimensional Bell polynomials | Vector partitions | Multi-dimensional Faà di Bruno formulae | MATHEMATICS, APPLIED | Multi-dimensional Faa di Bruno formulae | Partitions | Polynomials | Algorithms | Mathematical analysis | Combinatorial analysis | Mathematics | Computer Science | Classical Analysis and ODEs | Discrete Mathematics

Multi-dimensional Bell polynomials | Vector partitions | Multi-dimensional Faà di Bruno formulae | MATHEMATICS, APPLIED | Multi-dimensional Faa di Bruno formulae | Partitions | Polynomials | Algorithms | Mathematical analysis | Combinatorial analysis | Mathematics | Computer Science | Classical Analysis and ODEs | Discrete Mathematics

Journal Article

Complex Analysis and Operator Theory, ISSN 1661-8254, 2/2016, Volume 10, Issue 2, pp. 409 - 435

We show that Faà di Bruno’s formula can play important roles in modular forms theory and in the study of differential operators of the form $$ \displaystyle...

Lagrange inversion formula | Operator Theory | Modular forms | Analysis | Mathematics, general | Eisenstein series | Mathematics | Faà di Bruno’s formula | Complex Variables | Functional Analysis | Classical Analysis and ODEs

Lagrange inversion formula | Operator Theory | Modular forms | Analysis | Mathematics, general | Eisenstein series | Mathematics | Faà di Bruno’s formula | Complex Variables | Functional Analysis | Classical Analysis and ODEs

Journal Article

Mediterranean Journal of Mathematics, ISSN 1660-5446, 10/2016, Volume 13, Issue 5, pp. 2795 - 2800

In the paper, the author finds an explicit formula for the Bell numbers in terms of the Lah numbers and the Stirling numbers of the second kind.

Lah number | Faà di Bruno formula | Secondary 11B75 | 26A24 | 33B10 | Mathematics | derivative | exponential function | Primary 11B73 | Bell number | Bell polynomial | Explicit formula | Mathematics, general | Stirling number of the second kind | MATHEMATICS | MATHEMATICS, APPLIED | Faa di Bruno formula

Lah number | Faà di Bruno formula | Secondary 11B75 | 26A24 | 33B10 | Mathematics | derivative | exponential function | Primary 11B73 | Bell number | Bell polynomial | Explicit formula | Mathematics, general | Stirling number of the second kind | MATHEMATICS | MATHEMATICS, APPLIED | Faa di Bruno formula

Journal Article

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, ISSN 1578-7303, 1/2019, Volume 113, Issue 1, pp. 1 - 9

In the paper, the authors discuss the Bell polynomials and a sequence of polynomials applied to the theory of differential equations. Concretely speaking, the...

Faà di Bruno formula | 11C08 | Generating function | Secondary 11B73 | Differential equation | Theoretical, Mathematical and Computational Physics | 33B10 | Mathematics | Bell polynomial | Explicit formula | Primary 11B83 | Derivative | Mathematics, general | 26A06 | Applications of Mathematics | Identity | 26A09 | Stirling number | INEQUALITIES | Faa di Bruno formula | TERMS | DIAGONAL RECURRENCE RELATIONS | STIRLING NUMBERS | MATHEMATICS | 2ND KIND | Functions (mathematics) | Polynomials | Mathematical analysis | Combinatorial analysis | Differential equations | Identities

Faà di Bruno formula | 11C08 | Generating function | Secondary 11B73 | Differential equation | Theoretical, Mathematical and Computational Physics | 33B10 | Mathematics | Bell polynomial | Explicit formula | Primary 11B83 | Derivative | Mathematics, general | 26A06 | Applications of Mathematics | Identity | 26A09 | Stirling number | INEQUALITIES | Faa di Bruno formula | TERMS | DIAGONAL RECURRENCE RELATIONS | STIRLING NUMBERS | MATHEMATICS | 2ND KIND | Functions (mathematics) | Polynomials | Mathematical analysis | Combinatorial analysis | Differential equations | Identities

Journal Article

Journal of Pure and Applied Algebra, ISSN 0022-4049, 10/2019, Volume 223, Issue 10, pp. 4191 - 4225

Differential categories were introduced by Blute, Cockett, and Seely as categorical models of differential linear logic and have since led to abstract...

Faà di Bruno formula | Hurwitz series rings | Differential categories | Differential algebras | FORMS | MATHEMATICS | MATHEMATICS, APPLIED | MODELS | Faa di Bruno formula | Analysis | Algebra | Mathematics - Category Theory

Faà di Bruno formula | Hurwitz series rings | Differential categories | Differential algebras | FORMS | MATHEMATICS | MATHEMATICS, APPLIED | MODELS | Faa di Bruno formula | Analysis | Algebra | Mathematics - Category Theory

Journal Article

Theoretical Population Biology, ISSN 0040-5809, 06/2008, Volume 73, Issue 4, pp. 543 - 551

We show that the formula of Faà di Bruno for the derivative of a composite function gives, in special cases, the sampling distributions in population genetics...

Partition | Faà di Bruno's formula | Ewens sampling formula | Derivative | Probability | Taylor series | Composite function | Fisher logarthimic series | Negative-binomial | Compound sampling models | NEUTRAL ALLELES | Faa di Bruno's formula | probability | SAMPLING THEORY | derivative | MODEL | negative-binomial | EVOLUTIONARY BIOLOGY | partition | composite function | GENETICS & HEREDITY | ECOLOGY | compound sampling models | Genetics, Population | Binomial Distribution | Sample Size | Models, Genetic | Population genetics

Partition | Faà di Bruno's formula | Ewens sampling formula | Derivative | Probability | Taylor series | Composite function | Fisher logarthimic series | Negative-binomial | Compound sampling models | NEUTRAL ALLELES | Faa di Bruno's formula | probability | SAMPLING THEORY | derivative | MODEL | negative-binomial | EVOLUTIONARY BIOLOGY | partition | composite function | GENETICS & HEREDITY | ECOLOGY | compound sampling models | Genetics, Population | Binomial Distribution | Sample Size | Models, Genetic | Population genetics

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2017, Volume 2017, Issue 1, pp. 1 - 8

In the paper, by the Faà di Bruno formula, the authors establish two explicit formulas for the Motzkin numbers, the generalized Motzkin numbers, and the...

Faà di Bruno formula | 05A20 | Motzkin number | generating function | Mathematics | restricted hexagonal number | 11B37 | explicit formula | 11B83 | Analysis | generalized Motzkin number | Catalan number | Bell polynomial of the second kind | Mathematics, general | 05A19 | Applications of Mathematics | 05A15 | MATHEMATICS | MATHEMATICS, APPLIED | Faa di Bruno formula | CATALAN NUMBERS | Inequalities | Research

Faà di Bruno formula | 05A20 | Motzkin number | generating function | Mathematics | restricted hexagonal number | 11B37 | explicit formula | 11B83 | Analysis | generalized Motzkin number | Catalan number | Bell polynomial of the second kind | Mathematics, general | 05A19 | Applications of Mathematics | 05A15 | MATHEMATICS | MATHEMATICS, APPLIED | Faa di Bruno formula | CATALAN NUMBERS | Inequalities | Research

Journal Article

Discrete Applied Mathematics, ISSN 0166-218X, 2005, Volume 148, Issue 3, pp. 246 - 255

The coefficients of g ( s ) in expanding the rth derivative of the composite function g ∘ f by Faà di Bruno's formula, is determined by a Diophantine linear...

Diophantine equations with lower and upper bounds | Knapsack systems | Integer partitions | Lattices | Derivatives of composite functions | derivatives of composite functions | MATHEMATICS, APPLIED | diophantine equations with lower and upper bounds | FAA | COMPOSITE FUNCTIONS | lattices | DERIVATIVES | integer partitions

Diophantine equations with lower and upper bounds | Knapsack systems | Integer partitions | Lattices | Derivatives of composite functions | derivatives of composite functions | MATHEMATICS, APPLIED | diophantine equations with lower and upper bounds | FAA | COMPOSITE FUNCTIONS | lattices | DERIVATIVES | integer partitions

Journal Article

Journal of Computational and Applied Mathematics, ISSN 0377-0427, 04/2014, Volume 260, pp. 201 - 207

Guo and Qi (2013) posed a problem asking to determine the coefficients ak,i−1 for 1≤i≤k such that 1/(1−e−t)k=1+∑i=1kak,i−1(1/(et−1))(i−1). The authors answer...

Fubini number | Faà di Bruno’s formula | Apostol–Bernoulli number | Exponential function | Stirling number | Explicit expression | Q-ANALOG | MATHEMATICS, APPLIED | Apostol-Bernoulli number | Faa di Bruno's formula | APOSTOL-BERNOULLI | DI-BRUNOS FORMULA | Exponential functions | Mathematical models | Computation | Mathematical analysis | Combinatorial analysis

Fubini number | Faà di Bruno’s formula | Apostol–Bernoulli number | Exponential function | Stirling number | Explicit expression | Q-ANALOG | MATHEMATICS, APPLIED | Apostol-Bernoulli number | Faa di Bruno's formula | APOSTOL-BERNOULLI | DI-BRUNOS FORMULA | Exponential functions | Mathematical models | Computation | Mathematical analysis | Combinatorial analysis

Journal Article

Topology and its Applications, ISSN 0166-8641, 02/2018, Volume 235, pp. 375 - 427

In this paper, we consider abelian functor calculus, the calculus of functors of abelian categories established by the second author and McCarthy. We carefully...

Abelian categories | Faà di Bruno formula | Chain rule | Functor calculus | Cartesian differential categories | MATHEMATICS | MATHEMATICS, APPLIED | Faadi Bruno formula

Abelian categories | Faà di Bruno formula | Chain rule | Functor calculus | Cartesian differential categories | MATHEMATICS | MATHEMATICS, APPLIED | Faadi Bruno formula

Journal Article

Complex Variables and Elliptic Equations, ISSN 1747-6933, 02/2008, Volume 53, Issue 2, pp. 159 - 175

We study the singular behaviour of k-th angular derivatives of analytic functions in the unit disk in the complex plane ℂ and positive harmonic functions in...

Angular derivatives | Faà di Bruno's formula | Positive harmonic functions | MATHEMATICS | Faa di Bruno's formula

Angular derivatives | Faà di Bruno's formula | Positive harmonic functions | MATHEMATICS | Faa di Bruno's formula

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 2003, Volume 16, Issue 6, pp. 975 - 979

A short proof of the generalized Faa di Bruno formula is given and an explicit parametrization of the set of indices involved in the coefficient of a specific...

Differential calculus | Composite functions | Partial derivatives | composite functions | differential calculus | MATHEMATICS, APPLIED | partial derivatives | DERIVATIVES | BRUNO,FAA,DI FORMULA

Differential calculus | Composite functions | Partial derivatives | composite functions | differential calculus | MATHEMATICS, APPLIED | partial derivatives | DERIVATIVES | BRUNO,FAA,DI FORMULA

Journal Article

The ANZIAM Journal, ISSN 1446-1811, 1/2007, Volume 48, Issue 3, pp. 327 - 341

The Faa di Bruno formule for higher-order derivatives of a composite function are important in analysis for a variety of applications. There is a substantial...

Faà di Bruno formula | Multivariate Taylor series | Integral remainder term | Differentiation theory | Multivariate composite functions | POLYNOMIALS | MATHEMATICS, APPLIED | multivariate composite functions | CHAIN RULE | Faa di Bruno formula | multivariate Taylor series | integral remainder term | PARTITIONS | DIFFERENTIATION | differentiation theory

Faà di Bruno formula | Multivariate Taylor series | Integral remainder term | Differentiation theory | Multivariate composite functions | POLYNOMIALS | MATHEMATICS, APPLIED | multivariate composite functions | CHAIN RULE | Faa di Bruno formula | multivariate Taylor series | integral remainder term | PARTITIONS | DIFFERENTIATION | differentiation theory

Journal Article

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