Multivariate Lagrange Formula

• Multivariate Lagrange formula is a generalization of univariate inversion and interpolation formulas to multiple variables, enabling explicit inversion of analytic systems.
• It integrates analytic, algebraic, and combinatorial methods to construct interpolating polynomials and extract coefficients in high-dimensional settings.
• The approach leverages Jacobian determinants, Fourier analytic proofs, and combinatorial species theory to ensure stability and efficiency in diverse applications.

The multivariate Lagrange formula is a central result in the theory of analytic and formal power series, polynomial interpolation, combinatorial species, and analytic combinatorics. It generalizes the classical univariate Lagrange inversion and interpolation formulas to several variables and, in even more abstract versions, to function spaces and noncommutative settings. This framework enables explicit inversion of systems of analytic equations, construction of interpolating polynomials in higher dimensions, and determination of combinatorial structure constants in multivariate settings. The multivariate Lagrange formula appears in analytic, algebraic, combinatorial, and geometric contexts, with several distinct but related incarnations, including Lagrange–Good inversion, polynomial interpolation over commutative and noncommutative fields, multivariate variational calculus, and combinatorial species theory.

1. Classical and Multivariate Lagrange–Good Inversion

The classical Lagrange inversion formula reconstructs the coefficients of $h(w)$ defined implicitly by $h(w) = w g(h(w))$ for a power series $g$. In the multivariate generalization, one considers analytic (or formal power series) maps $g = (g_1, ..., g_n)$ in $n$ variables and solves the vector functional equation

$h_i(w) = w_i g_i(h(w)), \qquad i = 1, ..., n,$

which can be written in vector notation as $h(w) = \operatorname{diag}(w) \, g(h(w))$ (Hassairi et al., 22 Dec 2025).

The multivariate Lagrange–Good formula provides an explicit coefficient-extraction relation:

$$[w^k]\,g_0(h(w)) = [z^k]\left(g_0(z) \, g(z)^k\,D(g)(z)\right),$$

where $D(g)(z)$ is the Jacobian determinant

$$D(g)(z) = \det\left(I_n - \left[\frac{z_i}{g_i(z)} \frac{\partial g_i}{\partial z_j}(z)\right]_{i,j}\right)$$

and $g_0$ is any analytic function (Hassairi et al., 22 Dec 2025). This formula enables inversion of systems of equations and extraction of coefficients in composite analytic maps.

The algebraic and analytic criteria for the validity of the formula require $g=(g_1,\ldots,g_n)$ and $g_0$ to be analytic in a neighborhood of $0 \in \mathbb{C}^n$, with $g(0)\neq 0$ and nonvanishing Jacobian determinant at $0$ (Hassairi et al., 22 Dec 2025). The univariate case $n=1$ recovers the classical Lagrange expansion (Nguyen, 2023).

2. Polynomial Interpolation: Multivariate and Quaternionic Contexts

The multivariate Lagrange formula also arises in polynomial interpolation where, given a set of distinct nodes $\Theta = \{x_0, ..., x_n\} \subset H$ (for $H$ a field or skew-field), one constructs unique basis polynomials $\ell_j(x)$ of degree $\leq n$ satisfying $\ell_j(x_k) = \delta_{jk}$. For quaternionic variables, two canonical approaches are provided (Waldron, 2020):

Construction I (Symmetrized vanishing polynomials):

$$\ell_j(x) = \frac{1}{|A|!} \sum_{\sigma \in S_{|A|}} \left(\prod_{i} (x - x_{\sigma(i)}) \right) \bigl(p_j(x_j)\bigr)^{-1},$$

where $A = \Theta \setminus \{x_j\}$ and $p_j(x_j)$ is invertible for generic nodes.

Construction II (Symmetrized univariate formula):

$$\ell_j(x) = \frac{1}{n!} \sum_{\substack{\sigma \in S_{n+1}\ \sigma(j)=j}} \prod_{k\neq j} (x - x_{\sigma(k)}) (x_j - x_{\sigma(k)})^{-1}.$$

The associated interpolation operator is $L_\Theta f(x) = \sum_j \ell_j(x) f(x_j)$. These constructions ensure symmetry, translation-invariance, and continuity in the node positions (Waldron, 2020).

In commutative multivariate settings such as $\mathbb{C}^2$ and $\mathbb{R}^d$, explicit interpolation schemes exist, such as those for Leja points or checkerboard nodes (Irigoyen, 2014, Cao et al., 2021), and the unisolvency of such schemes is based on Vandermonde-determinant arguments or dimension counts.

3. Jacobian Structure and Analytic Proofs

The Jacobian determinant $D(g)(z)$ is essential for the validity of the multivariate inversion formula. In the analytic approach pioneered by Good and further streamlined by Dirac–Fourier methods, the Jacobian factor arises naturally as the change-of-variables determinant in the transformation of an integral representation (Nguyen, 2023). Specifically, for the system $g_i(x) = x_i f_i(g(x))$, the Jacobian is $\det(\delta_{ij} - x_i \partial_j f_i(g(x)))$. Analytically, this factor ensures unique root selection and proper coefficient extraction.

Recent proofs replace several pages of classical residue calculus with direct applications of the Fourier representation of the Dirac delta:

$$\delta(u) = \frac{1}{(2\pi)^n} \int_{\mathbb{R}^n} e^{-i \xi \cdot u} d\xi,$$

with combinatorial coefficients emerging only from the multinomial expansion (Nguyen, 2023). This analytic machinery clarifies the combinatorial validity of the inversion formula in both formal and analytic contexts.

4. Combinatorial and Infinite-dimensional Extensions

The multivariate Lagrange–Good formula generalizes further to settings with countably or uncountably many variables, as formalized in the language of combinatorial species. In this abstract setting, a formal power series is encoded as a sequence of symmetric kernels $F_n:\mathbb{X}^n \to \mathbb{C}$, and interpolation, inversion, and coefficient extraction are achieved via a Fredholm determinant acting on kernel operators:

$$\Psi_n(q_1,\ldots,q_n) = [\text{const in }z] \, \frac{\delta^n}{\delta z(q_1) \cdots \delta z(q_n)}\left\{ \Phi(z) \exp\left( \sum_{i=1}^n A(q_i;z) \right) \det(\mathrm{Id} - \mathbb{A}_z) \right\},$$

where $\mathbb{A}_z$ is an integral operator built from variational derivatives of the underlying kernel $A$ (Jansen et al., 2020).

This setting encompasses inversion results for measures on color spaces, analytic combinatorics on graphs, as well as expansions for statistical mechanics and field theory (e.g., tree expansions, cluster/virial expansions).

For finite sets, the Fredholm determinant reduces to the classical, finite-dimensional Jacobian determinant and standard Lagrange–Good forms (Jansen et al., 2020).

5. Explicit Formulas and Special Constructions

In applied settings, explicit forms of the multivariate Lagrange formula are valuable for practical computation and error analysis. For bidimensional Leja sequences in $\mathbb{C}^2$, the fundamental Lagrange interpolation polynomial (FLIP) at a node $H_j$ can be written as a quotient of Vandermonde determinants or, in certain cases, as a sum of explicit one-variable products (Irigoyen, 2014):

$$\ell_j^{(N)}(z,w) = \frac{\det[e_i(H_k)]_{1 \leq i,k \leq N, \, H_j \mapsto (z,w)}}{\det[e_i(H_k)]_{1 \leq i,k \leq N}},$$

with indices ordered by graded lexicographic degree.

For checkerboard node systems in $\mathbb{R}^2$, bivariate Lagrange polynomials are constructed by combining univariate orthogonal polynomial kernels and Christoffel–Darboux formulas, with uniqueness established in quotient spaces of vanishing polynomials (Cao et al., 2021).

Quaternionic interpolation leverages symmetrization and permutation-averaging to restore symmetry lost due to noncommutativity, with precise existence and uniqueness theorems governing when interpolation schemes are valid (Waldron, 2020).

6. Geometric Properties and Stability

Multivariate Lagrange interpolation schemes exhibit key geometric invariances: translation invariance ($\ell_j(x+a ; \Theta+a) = \ell_j(x ; \Theta)$), symmetry in node labelling, and, in typical generic configurations, localization of zeros ($\ell_j$ vanishes exactly on the set $\Theta\setminus \{x_j\}$ in quaternionic contexts) (Waldron, 2020). Uniform boundedness and Lebesgue constant growth are critical for numerical stability and convergence. For Leja sequences in the unit disk and bidisk, explicit constants and polynomial growth rates are established; e.g., the Lebesgue constant in one dimension is $O(N)$, while in two variables it is $O(N^{3/2})$ (Irigoyen, 2014). Such stability guarantees underpin the reliability of these schemes in theoretical and applied computation.

7. Applications and Connections

The multivariate Lagrange formula is a foundational tool in analytic combinatorics (enumeration of trees, plane maps), solution of functional equations in statistical mechanics (density/activity inversions, cluster expansions), field theory (Gallavotti trees, KAM series), and the explicit construction of discrete measures for natural exponential families supported on $\mathbb{N}^n$ (Hassairi et al., 22 Dec 2025, Jansen et al., 2020). Its group-theoretic interpretation in the study of variance functions associates the analytic combinatorial machinery with algebraic structures such as $GL(n+1, \mathbb{R})$ acting on cumulant and variance maps (Hassairi et al., 22 Dec 2025).

Lagrange interpolation schemes in noncommutative (quaternionic) settings connect to geometric function theory, while multivariate polynomial interpolation underpins high-dimensional approximation, numerical analysis, and signal processing. The Fredholm determinant extension and the explicit cancellation of cycle-rooted components clarify the interplay between algebraic determinants and tree expansions in combinatorial species (Jansen et al., 2020).

In summary, the multivariate Lagrange formula synthesizes analytic, algebraic, geometric, and combinatorial methodologies, providing explicit, universal tools for inversion, interpolation, and coefficient extraction in multivariate and even infinite-dimensional analytic and combinatorial settings.