Geodesic
Some generalizations of the Faa Di Bruno formula
Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 180-193

Voir la notice de l'article provenant de la source Math-Net.Ru

The focus of the article is the classical Faa Di Bruno formula for computing higher-order derivatives of a complex function $F(u(x))$. Here is a version of the proof of this formula. Then we prove a generalization of the Faa Di Bruno formula to the case of a complex function with an inner function $u(x,y)$ depending on two independent variables. The paper presents a formula for the $n$-th derivative of a complex function, when the argument of the outer function is a vector with an arbitrary number of components (functions of one variable). The article also considers examples of finding higher-order derivatives, illustrating both the classical Faa Di Bruno formula and its generalizations.
Mots-clés : Faa Di Bruno's formula
Keywords: $n$-th derivative of complex functions of several variables, generalizations of Faa Di Bruno's formula for these functions, Newton's binomial and polynomial formulas. 
P. N. Sorokin. Some generalizations of the Faa Di Bruno formula. Čebyševskij sbornik, Tome 24 (2023) no. 5, pp. 180-193. http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a11/
@article{CHEB_2023_24_5_a11,
     author = {P. N. Sorokin},
     title = {Some generalizations of the {Faa} {Di} {Bruno} formula},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {180--193},
     year = {2023},
     volume = {24},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a11/}
}
TY  - JOUR
AU  - P. N. Sorokin
TI  - Some generalizations of the Faa Di Bruno formula
JO  - Čebyševskij sbornik
PY  - 2023
SP  - 180
EP  - 193
VL  - 24
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a11/
LA  - ru
ID  - CHEB_2023_24_5_a11
ER  - 
%0 Journal Article
%A P. N. Sorokin
%T Some generalizations of the Faa Di Bruno formula
%J Čebyševskij sbornik
%D 2023
%P 180-193
%V 24
%N 5
%U http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a11/
%G ru
%F CHEB_2023_24_5_a11

[1] Faá di Bruno F., “Sullo sviluppo delle funzione”, Annali di Scuenze Matematiche e Fisiche, 6 (1855), 479–480

[2] Faá di Bruno F., “Note sur un nouvelle formulae de calcul differentiel”, Quart. J. Math., 1 (1857), 359–360

[3] Mishkov R. L., “Generalization of the formula of Faá di Bruno for a composite function with a vector argument”, Internat. J. Math. Math. Sci., 24:7 (2000), 481–491 | DOI | MR | Zbl

[4] Roman S., “The formula of Faá di Bruno”, Amer. Math. Monthly, 87:10 (1980), 805–809 | DOI | MR | Zbl

[5] Dvoryaninov S. V., Silvanovich M. I., “On the Faá di Bruno formula for derivatives of a complex function”, Matematicheskoye obrazovaniye, 2009, no. 1(49), 22–26

[6] Arhipov G. I., Chubarikov V. N., Sadovnichiy V. A., Lectures on mathematical analysis, Drofa, M., 2003, 640 pp.

[7] Bell E. T., “Partition polynomials”, Ann. Math., 29 (1927), 38–46 | DOI | MR

[8] Comtet L., Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974 | MR | Zbl

[9] Johnson W. P., “The curious history of Faá di Bruno's formula”, Amer. Math. Monthly, 109 (2002), 217–234 | MR | Zbl

[10] Chubarikov V. N., “A generalized Binomial theorem and a summation formulas”, Chebyshevskii Sbornik, 21:4 (2020), 270–301 | DOI | MR | Zbl

[11] Constantine G. M., Savits T. H., “A multivariate Faá di Bruno formula with applications”, Trans. Amer. Math. Soc., 348:2 (1996), 503–520 | DOI | MR | Zbl

[12] Shabat A. B., Efendiev M. Kh., “Applications of the Faá di Bruno formula”, Ufimskiy matematicheskiy zhurnal, 9:3 (2017), 132–137 | MR | Zbl

[13] Demidovich B. P., Sbornik zadach i uprazhneniy po matematicheskomu analizu, Uchebnoye posobiye, 14 izdaniye, ispr., Izd-vo Moskovskogo universiteta, M., 1998, 624 pp.

[14] Frabetti A., Manchon D., Five interpretation of Faá di Bruno's formula, 2014, arXiv: 1402.5551 | MR

[15] Craik A. D. D., “Prehistory of Faá di Bruno's formula”, Amer. Math. Monthly, 112:2 (2005), 119–130 | MR | Zbl

[16] Arbogast L. F. A., Du Calcul des Dérivations, Levrault, Strasbourg, 1800