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

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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.
Keywords: Faa Di Bruno's formula, $n$-th derivative of complex functions of several variables, generalizations of Faa Di Bruno's formula for these functions, Newton's binomial and polynomial formulas. 
@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},
     publisher = {mathdoc},
     volume = {24},
     number = {5},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2023_24_5_a11/}
}
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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/