Expansion of implicit functions into formal power series in terms of partial Bell polynomials
DOI:
https://doi.org/10.13069/jacodesmath.v13i2.397Keywords:
Implicit function, Formal power series, Higher derivatives, Series reversion, Bell polynomials, Multivariable Stirling polynomialsAbstract
Starting from the representation of a function \( f(x,y) \) as a formal power series with Taylor coefficients \( f_{m,n} \), a formal series is set up for the implicit function \( y = y(x) \) so that \( f(x,y) = 0 \) and the coefficients of the series for \( y \) depend exclusively on the \( f_{m,n} \). The solution to this problem provided here relies on using partial Bell polynomials and their inverse companions. Some examples and applications are discussed.
Accepted: 15 March 2026
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References
L. Comtet, Polynômes de Bell et formule explicite des dérivées successives d’une fonction implicite, C. R. Acad. Sci. Paris Série A, t. 267 (1968) 457–460.
L. Comtet, Advanced combinatorics. The art of finite and infinite expansions, rev. and enlarged edition. Reidel, Dordrecht (Holland) 1974.
L. Comtet, M. Fiolet, Sur les dérivées successives d’une fonction implicite, C. R. Acad. Sci. Paris Série A t. 278 (1974) 249–251.
W. P. Johnson, Some problems in differentiation, arXiv:1811.12293, version 1, 29 Nov 2018.
J. H. Lambert, Observationes variae in mathesin puram. Acta Helvetica 3 (1758) 128–168 (§38 Opera Mathematica 1 16–51, Orell Füssli Verlag, Zürich 1946).
J. Riordan, Introduction to combinatorial analysis, New York, 1958 reprint Dover Publ. Inc. Mineola New York 2002.
A. Schreiber, Multivariate Stirling polynomials of the first and second kind, Discrete Math. 338 (2015) 2462–2484.
A. Schreiber, Inverse relations and reciprocity laws involving partial Bell polynomials and related extensions, Enumer. Combin. Appl. 1(1) (2021) Article S2R3.
A. Schreiber, Stirling polynomials in several indeterminates, Logos Verlag Berlin 2021.
J. Stillwell, Eisenstein’s footnote, The Math. Intelligencer 17(2) (1995) 58–62.
T. Wilde, Implicit higher derivatives, and a formula of Comtet and Fiolet, arXiv:0805.2674, version 1, 17 May 2008.
S. Zemel, The Combinatorics of higher derivatives of implicit functions, Monatsh. Math. 188(4) (2019) 765–784. Preprint: https://arxiv.org/abs/1711.07501.
