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Journal articles on the topic 'Faa di Bruno's formula'

Author: Grafiati

Published: 28 May 2022

Last updated: 31 July 2025

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1

Gzyl, Henryk. "Multidimensional extension of Faa di Bruno's formula." Journal of Mathematical Analysis and Applications 116, no. 2 (1986): 450–55. http://dx.doi.org/10.1016/s0022-247x(86)80009-9.

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2

Mishkov, Rumen L. "Generalization of the formula of Faa di Bruno for a composite function with a vector argument." International Journal of Mathematics and Mathematical Sciences 24, no. 7 (2000): 481–91. http://dx.doi.org/10.1155/s0161171200002970.

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The paper presents a new explicit formula for thenth total derivative of a composite function with a vector argument. The well-known formula of Faa di Bruno gives an expression for thenth derivative of a composite function with a scalar argument. The formula proposed represents a straightforward generalization of Faa di Bruno's formula and gives an explicit expression for thenth total derivative of a composite function when the argument is a vector with an arbitrary number of components. In this sense, the formula of Faa di Bruno is its special case. The mathematical tools used include differe

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3

Johnson, Warren P. "The Curious History of Faa di Bruno's Formula." American Mathematical Monthly 109, no. 3 (2002): 217. http://dx.doi.org/10.2307/2695352.

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4

Corcino, Roberto Bagsarsa, Charles Montero, Maribeth Montero, and Jay Ontolan. "The r-Dowling Numbers and Matrices Containing r-Whitney Numbers of the Second Kind and Lah Numbers." European Journal of Pure and Applied Mathematics 12, no. 3 (2019): 1122–37. http://dx.doi.org/10.29020/nybg.ejpam.v12i3.3494.

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This paper derives another form of explicit formula for $(r,\beta)$-Bell numbers using the Faa di Bruno's formula and certain identity of Bell polynomials of the second kind. This formula is expressed in terms of the $r$-Whitney numbers of the second kind and the ordinary Lah numbers. As a consequence, a relation between $(r,\beta)$-Bell numbers and the sums of row entries of the product of two matrices containing the $r$-Whitney numbers of the second kind and the ordinary Lah numbers is established. Moreover, a $q$-analogue of the explicit formula is obtained.

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5

Dawid, A. P., S. Kotz, N. L. Johnson, and C. B. Read. "Encyclopedia of Statistical Sciences, Vol. 3. Faa di Bruno's Formula- Hypothesis Testing." Biometrics 41, no. 1 (1985): 341. http://dx.doi.org/10.2307/2530670.

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6

FERGER, Dietmar. "Moment equalities for sums of random variables via integer partitions and Faa di Bruno's formula." TURKISH JOURNAL OF MATHEMATICS 38 (2014): 558–75. http://dx.doi.org/10.3906/mat-1301-6.

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7

Jha, Sumit Kumar. "A formula for the number of non-negative integer solutions of a1x1 + a2x2 + ··· + amxm = n in terms of the partial Bell polynomials." Notes on Number Theory and Discrete Mathematics 27, no. 2 (2021): 64–69. http://dx.doi.org/10.7546/nntdm.2021.27.2.64-69.

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8

B, G. C. Keshan, and K. Wijerathna J. "Implementation and validation of a closed form formula for implied volatility." Indian Journal of Science and Technology 13, no. 38 (2020): 4064–72. https://doi.org/10.17485/IJST/v13i38.1301.

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Abstract <strong>Introduction:</strong> Implied volatility is one of the most commonly used estimator to predict future stock market behaviours. Accordingly, millions of option prices are used to compute the implied volatility in the stock market frequently. Traditionally, this is calculated by inverting Black-Scholes option prices with iterative numerical methods but this associates high computational cost. <strong>Objectives:</strong> This research was mainly focused on implementing and validating an explicit closed-form formula for the implied volatility by using market obser

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9

沈, 萍. "Application of Faà di Bruno’s Formula in Identity and Counting." Pure Mathematics 12, no. 12 (2022): 2231–38. http://dx.doi.org/10.12677/pm.2022.1212240.

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10

Craik, Alex D. D. "Prehistory of Faà di Bruno's Formula." American Mathematical Monthly 112, no. 2 (2005): 119–30. http://dx.doi.org/10.1080/00029890.2005.11920176.

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11

Craik, Alex D. D. "Prehistory of FaÃÂ di Bruno's Formula." American Mathematical Monthly 112, no. 2 (2005): 119. http://dx.doi.org/10.2307/30037410.

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12

Encinas, L. Hernández, A. Martín del Rey, and J. Muñoz Masqué. "Faà di Bruno's formula, lattices, and partitions." Discrete Applied Mathematics 148, no. 3 (2005): 246–55. http://dx.doi.org/10.1016/j.dam.2005.02.009.

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13

Johnson, Warren P. "Aq-Analogue of Faà di Bruno's Formula." Journal of Combinatorial Theory, Series A 76, no. 2 (1996): 305–14. http://dx.doi.org/10.1006/jcta.1996.0107.

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14

Johnson, Warren P. "The Curious History of Faà di Bruno's Formula." American Mathematical Monthly 109, no. 3 (2002): 217–34. http://dx.doi.org/10.1080/00029890.2002.11919857.

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15

Johnston, Samuel G. G., and Joscha Prochno. "Faà di Bruno's formula and inversion of power series." Advances in Mathematics 395 (February 2022): 108080. http://dx.doi.org/10.1016/j.aim.2021.108080.

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16

Encinas, L. Hernández, and J. Muñoz Masqué. "A short proof of the generalized Faà di bruno's formula." Applied Mathematics Letters 16, no. 6 (2003): 975–79. http://dx.doi.org/10.1016/s0893-9659(03)90026-7.

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17

Chou, W. S., Leetsch C. Hsu, and Peter J. S. Shiue. "Application of Faà di Bruno's formula in characterization of inverse relations." Journal of Computational and Applied Mathematics 190, no. 1-2 (2006): 151–69. http://dx.doi.org/10.1016/j.cam.2004.12.041.

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18

Constantine, G. M., and T. H. Savits. "A Multivariate Faa di Bruno Formula with Applications." Transactions of the American Mathematical Society 348, no. 2 (1996): 503–20. http://dx.doi.org/10.1090/s0002-9947-96-01501-2.

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19

Shabat, Alexey Borisovich, and Magomed Khochalaevich Efendiev. "On applications of Faà-di-Bruno formula." Ufimskii Matematicheskii Zhurnal 9, no. 3 (2017): 131–36. http://dx.doi.org/10.13108/2017-9-3-131.

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20

Schmee, Josef, Samuel Kotz, and Norman L. Johnson. "Encyclopedia of Statistical Sciences (Vol. 3: Faà di Bruno's Formula-Hypothesis Testing)." Technometrics 27, no. 2 (1985): 210. http://dx.doi.org/10.2307/1268774.

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21

Pan, Y., and M. Wang. "On higher order angular derivatives – an application of Faà di Bruno's formula." Complex Variables and Elliptic Equations 53, no. 2 (2008): 159–75. http://dx.doi.org/10.1080/17476930701669548.

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22

Schmee, Josef. "Encyclopedia of Statistical Sciences (Vol. 3: Fa di Bruno's Formula–Hypothesis Testing)." Technometrics 27, no. 2 (1985): 210. http://dx.doi.org/10.1080/00401706.1985.10488043.

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23

Ponomarenko, Vadim. "Faà di Bruno's formula and nonhyperbolic fixed points of one-dimensional maps." International Journal of Mathematics and Mathematical Sciences 2004, no. 29 (2004): 1543–49. http://dx.doi.org/10.1155/s0161171204306253.

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Fixed-point theory of one-dimensional maps ofℝdoes not completely address the issue of nonhyperbolic fixed points. This note generalizes the existing tests to completely classify all such fixed points. To do this, a family of operators are exhibited that are analogous to generalizations of the Schwarzian derivative. In addition, a family of functionsfare exhibited such that the Maclaurin series off(f(x))andxare identical.

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24

Di Nardo, Elvira, and Giuseppe Guarino. "kStatistics: Unbiased Estimates of Joint Cumulant Products from the Multivariate Faà Di Bruno's Formula." R Journal 14, no. 2 (2022): 209–29. http://dx.doi.org/10.32614/rj-2022-033.

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25

Singh, Jitender. "Sums of Products Involving Power Sums of φ(n) Integers". Journal of Numbers 2014 (27 лютого 2014): 1–6. http://dx.doi.org/10.1155/2014/158351.

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A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums Ψk(x,n):=∑d|n‍μ(d)dkSkx/d, n∈ℤ+ which are defined via the Möbius function μ and the usual power sum Sk(x) of a real or complex variable x. The power sum Sk(x) is expressible in terms of th

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26

Ben Zitoun, Feyed, and Yves Cherruault. "A Taylor expansion approach using Faà di Bruno's formula for solving nonlinear integral equations of the second and third kind." Kybernetes 38, no. 5 (2009): 800–818. http://dx.doi.org/10.1108/03684920910962687.

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27

Sarsengeldin, Merey, and Stanislav Kharina. "Method of the integral error functions for the solution of the one- and two-phase Stefan problems and its application." Filomat 31, no. 4 (2017): 1017–29. http://dx.doi.org/10.2298/fil1704017s.

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The analytical solutions of the one- and two-phase Stefan problems are found in the form of series containing linear combinations of the integral error functions which satisfy a priori the heat equation. The unknown coefficients are defined from the initial and boundary conditions by the comparison of the like power terms of the series using the Faa di Bruno formula. The convergence of the series for the temperature and for the free boundary is proved. The approximate solution is found using the replacement of series by the corresponding finite sums and the collocation method. The presented te

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28

López, Ronald Orozco. "Solution of the equation $y^{\prime}=f(y)$ and Bell Polynomials." Communications in Mathematics Volume 31 (2023), Issue 1 (November 11, 2022). http://dx.doi.org/10.46298/cm.10278.

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In this paper we use Faa di Bruno's formula to associate Bell polynomial values to differential equations of the form $y^{\prime}=f(y)$. That is, we use partial Bell polynomials to represent the solution of such an equation and use the solution to compute special values of partial Bell polynomials.

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29

Kumar, Jatin, Sunita, and Anil Saini. "UTILIZATION OF DIFFERENTIAL TRANSFORMATION METHOD TO NONLINEAR THREE-POINT SINGULAR BOUNDARY VALUE PROBLEMS." ShodhKosh: Journal of Visual and Performing Arts 5, no. 1 (2024). http://dx.doi.org/10.29121/shodhkosh.v5.i1.2024.2799.

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This paper explores the application of the Differential Transformation Method (DTM) to nonlinear three-point singular boundary value problems (SBVPs). The method incorporates "Faa di Bruno's formula," which utilizes "partial normal ring polynomials," to manage the nonlinearity of the equations. The paper presents error analysis results, comparing them with those obtained through other methods such as the reproducing kernel Hilbert space method (RKHSM). Numerical examples demonstrate the efficacy and reliability of the DTM in providing accurate solutions to these complex problems.

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30

Evers, Michael P., and Markus Kontny. "A Generalized Representation of Faà Di Bruno'S Formula Using Multivariate and Matrix‐Valued Bell Polynomials." Mathematical Methods in the Applied Sciences, July 20, 2025. https://doi.org/10.1002/mma.11226.

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ABSTRACTWe provide a generalization of Faà di Bruno's formula to represent the ‐th total derivative of the multivariate and vector‐valued composite . To this end, we make use of properties of the Kronecker product and the ‐th derivative of the left‐composite , which allow the use of a multivariate and matrix‐valued form of partial Bell polynomials to represent the generalized Faà di Bruno's formula. We further show that standard recurrence relations that hold for the univariate partial Bell polynomial also hold for the multivariate partial Bell polynomial under a simple transformation. We appl

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31

Hardy, Michael. "Combinatorics of Partial Derivatives." Electronic Journal of Combinatorics 13, no. 1 (2006). http://dx.doi.org/10.37236/1027.

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The natural forms of the Leibniz rule for the $k$th derivative of a product and of Faà di Bruno's formula for the $k$th derivative of a composition involve the differential operator $\partial^k/\partial x_1 \cdots \partial x_k$ rather than $d^k/dx^k$, with no assumptions about whether the variables $x_1,\dots,x_k$ are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables. Coefficients appearing in forms of these identities in which some variables are indistinguishable are just multiplicities of indistinguishable terms (in particular,

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32

Kosugi, Hiroki, Kanta Naito, and Spiridon Penev. "Semiparametric regression with localized Bregman divergence." Scandinavian Journal of Statistics, April 24, 2025. https://doi.org/10.1111/sjos.12789.

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AbstractThis paper focuses on semiparametric regression based on minimizing the localized Bregman divergence. A local parametric model derived from the framework of the generalized linear model with multiple covariates and a linear predictor is utilized. The parameter vector included in the model is estimated under localization. The asymptotic behavior of both the locally estimated parameter vector and the induced regression estimator is investigated. Theoretical comparisons of estimators by using the divergence risk measure are also addressed. Further generalization, including a multivariate

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33

Bokulić, Ana, Tajron Jurić, and Ivica Smolić. "Hexadecapole at the heart of nonlinear electromagnetic fields." Classical and Quantum Gravity, June 26, 2024. http://dx.doi.org/10.1088/1361-6382/ad5c34.

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Abstract In classical Maxwell's electromagnetism, monopole term of the electric field is proportional to $r^{-2}$, while higher order multipole terms, sourced by anisotropic sources, fall-off faster. However, in nonlinear electromagnetism even a spherically symmetric field has multipole-like contributions. We prove that the leading sub\-dominant term of the electric field, defined by nonlinear electromagnetic Lagrangian obeying Maxwellian weak field limit, in a static, spherically symmetric, asymptotically flat spacetime, is of the order $O(r^{-6})$ as $r \to \infty$. Moreover, using Lagrange

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34

Tavazza, Giuseppe. "Fractional derivatives of composite functions via an extension of Bell exponential polynomials." December 24, 2018. https://doi.org/10.5281/zenodo.2525740.

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In this work we present an extension of Bell exponential polynomials to fractional and negative indices. This extension allows to calculate the fractional derivatives and fractional integrals of composite functions using the Faà di Bruno formula

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35

Robinson, Thomas J. "Formal Calculus and Umbral Calculus." Electronic Journal of Combinatorics 17, no. 1 (2010). http://dx.doi.org/10.37236/367.

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We use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus. We begin by calculating the exponential generating function of the higher derivatives of a composite function, following a very short proof which naturally arose as a motivating computation related to a certain crucial "associativity" property of an important class of vertex operator algebras. Very similar (somewhat forgotten) proofs had appeared by the 19-th century, of course without any motivation related to vertex operator algebras. Using this for

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