Lagrange inversion of Taylor polynomials using Kronecker product notation

Roberto Leon-Gonzalez; Elnura Baiaman kyzy; Roberto Leon-Gonzalez; Elnura Baiaman kyzy

doi:10.3934/math.2026967

AIMS Mathematics

2026, Volume 11, Issue 8: 23995-24006. doi: 10.3934/math.2026967

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Research article

Lagrange inversion of Taylor polynomials using Kronecker product notation

Roberto Leon-Gonzalez ^{1
,
,},
Elnura Baiaman kyzy ²

1.
National Graduate Institute for Policy Studies (GRIPS), 7-22-1, Roppongi, Minato-Ku, Tokyo 106-8677, Japan
2.
HIAS, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan

Received: 15 April 2026 Revised: 01 July 2026 Accepted: 14 July 2026 Published: 05 August 2026
MSC : 13F25, 15A69, 40-08

This paper was concerned with the multivariate Lagrange-Good inversion of power series when written using Kronecker products and inverted with respect to a subset of the variables. This problem arose in the estimation of macroeconomic nonlinear dynamic stochastic general equilibrium (DSGE) models. It was shown that the problem of finding the coefficients of the inverse series became much simpler when using Kronecker products. An explicit recursive formula for the solution was derived from first principles, and, regardless of the number of variables, the formula had the same simplicity as in the univariate case. In this Kronecker product formulation, all the coefficients for a given order were contained in one matrix, and this matrix could be calculated with simple operations on other matrices of coefficients. The results of this paper provided a useful tool to estimate nonlinear DSGE models solved with a higher order approximation.
- DSGE,
- reversion of series,
- higher order inverse derivatives,
- perturbation methods,
- multinomial coefficient
Citation: Roberto Leon-Gonzalez, Elnura Baiaman kyzy. Lagrange inversion of Taylor polynomials using Kronecker product notation[J]. AIMS Mathematics, 2026, 11(8): 23995-24006. doi: 10.3934/math.2026967

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Abstract

This paper was concerned with the multivariate Lagrange-Good inversion of power series when written using Kronecker products and inverted with respect to a subset of the variables. This problem arose in the estimation of macroeconomic nonlinear dynamic stochastic general equilibrium (DSGE) models. It was shown that the problem of finding the coefficients of the inverse series became much simpler when using Kronecker products. An explicit recursive formula for the solution was derived from first principles, and, regardless of the number of variables, the formula had the same simplicity as in the univariate case. In this Kronecker product formulation, all the coefficients for a given order were contained in one matrix, and this matrix could be calculated with simple operations on other matrices of coefficients. The results of this paper provided a useful tool to estimate nonlinear DSGE models solved with a higher order approximation.

References

[1]	J. L. Lagrange, Mémoires de l'Académie royale des sciences, Belles-Lettres de Berlin, 1770.
[2]	E. Goursat, A course in mathematical analysis, I and II, part I, Ginn & Company, 1904.
[3]	K. Knopp, Theory and application of infinite series, Hafner, 1951.
[4]	T. M. Apostol, Calculating higher derivatives of inverses, Amer. Math. Mon., 107 (2018), 738–741. https://doi.org/10.1080/00029890.2000.12005264 doi: 10.1080/00029890.2000.12005264
[5]	W. P. Johnson, Combinatorics of higher derivatives of inverses, Amer. Math. Mon., 109 (2002), 273–277. https://doi.org/10.1080/00029890.2002.11919861 doi: 10.1080/00029890.2002.11919861
[6]	M. S. Floater, T. Lyche, Divided differences of inverse functions and partitions of a convex polygon, Math. Comput., 77 (2008), 2295–2308. https://doi.org/10.1090/S0025-5718-08-02144-3 doi: 10.1090/S0025-5718-08-02144-3
[7]	I. J. Good, Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes, Math. Proc. Cambridge Philos. Soc., 56 (1960), 367–380. https://doi.org/10.1017/S0305004100034666 doi: 10.1017/S0305004100034666
[8]	S. A. Joni, Lagrange inversion in higher dimensions and umbral operators, Linear Multilinear Algebra, 6 (1978), 111–121. https://doi.org/10.1080/03081087808817229 doi: 10.1080/03081087808817229
[9]	N. G. de Bruijn, The Lagrange-Good inversion formula and its application to integral equations, J. Math. Anal. Appl., 92 (1983), 397–409. https://doi.org/10.1016/0022-247X(83)90257-3 doi: 10.1016/0022-247X(83)90257-3
[10]	I. M. Gessel, A combinatorial proof of the multivariable lagrange inversion formula, J. Combin. Theory Ser. A, 45 (1987), 178–195. https://doi.org/10.1016/0097-3165(87)90013-6 doi: 10.1016/0097-3165(87)90013-6
[11]	B. Richmond, Multivariate Lagrange inversion, Algorithms Semin., 3504 (1998), 23–26.
[12]	E. A. Bender, L. B. Richmond, A multivariate Lagrange inversion formula for asymptotic calculations, Electron. J. Combin., 5 (1988), #R33. https://doi.org/10.37236/1371 doi: 10.37236/1371
[13]	L. H. Encinas, J. M. Masqué, A short proof of the generalized Faà di Bruno's formula, Appl. Math. Lett., 16 (2003), 975–979. https://doi.org/10.1016/S0893-9659(03)90026-7 doi: 10.1016/S0893-9659(03)90026-7
[14]	R. A. Horn, C. R. Johnson, Topics in matrix analysis, Cambridge University Press, 1991. https://doi.org/10.1017/CBO9780511840371
[15]	S. Adjemian, M. Juillard, F. Karamé, W. Mutschler, J. Pfeifer, M. Ratto, et al., Dynare: reference manual, version 6, Econpapers, 2024. Avaible from: https://econpapers.repec.org/paper/cpmdynare/080.htm.
[16]	S. Schmitt-Grohé, M. Uribe, Solving dynamic general equilibrium models using a second-order approximation to the policy function, J. Econ. Dyn. Control, 28 (2004), 755–775. https://doi.org/10.1016/S0165-1889(03)00043-5 doi: 10.1016/S0165-1889(03)00043-5
[17]	E. B. kyzy, R. Leon-Gonzalez, Estimation of nonlinear DSGE models through Laplace based solutions, J. Econ. Dyn. Control, 182 (2026), 105220. https://doi.org/10.1016/j.jedc.2025.105220 doi: 10.1016/j.jedc.2025.105220
[18]	Y. Okada, Surface deformation due to shear and tensile faults in a half-space, Bull. Seismol. Soc. Amer., 75 (1985), 1135–1154. https://doi.org/10.1785/BSSA0750041135 doi: 10.1785/BSSA0750041135
[19]	R. L. Parker, Geophysical inverse theory, Princeton University Press, 1994.
[20]	R. A. Beeler, How to count: an introduction to combinatorics and its applications, Springer, 2015. https://doi.org/10.1007/978-3-319-13844-2
[21]	G. H. Hardy, S. Ramanujan, Asymptotic formulaæ in combinatory analysis, Proc. London Math. Soc., 17 (1918), 75–115. https://doi.org/10.1112/plms/s2-17.1.75 doi: 10.1112/plms/s2-17.1.75
[22]	O. de Groot, Solving asset pricing models with stochastic volatility, J. Econ. Dyn. Control, 52 (2015), 308–321. https://doi.org/10.1016/j.jedc.2015.01.001 doi: 10.1016/j.jedc.2015.01.001
[23]	J. Fernández-Villaverde, O. Levintal, Solution methods for models with rare disasters, Quant. Econ., 9 (2018), 903–944. https://doi.org/10.3982/QE744 doi: 10.3982/QE744
[24]	O. Levintal, Fifth-order perturbation solution to DSGE models, J. Econ. Dyn. Control, 80 (2017), 1–16. https://doi.org/10.1016/j.jedc.2017.04.007 doi: 10.1016/j.jedc.2017.04.007
[25]	I. Stojmenovic, A. Zoghbi, Fast algorithms for generating integer partitions, Int. J. Comput. Math., 70 (1998), 319–332. https://doi.org/10.1080/00207169808804755 doi: 10.1080/00207169808804755

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