Generalized Euler Decomposition Formula

• Generalized Euler Decomposition Formula is a collection of results extending classical Euler identities to domains like convex polytopes, shuffle algebras, and Lie theory.
• It provides a unified framework for decomposing algebraic, combinatorial, and geometric structures, enabling the analysis of multiple zeta values, polynomial matrices, and q-analogs.
• These formulas underpin deeper structural insights in fields ranging from combinatorial topology to representation theory and continuum physics, facilitating computational verification and theoretical advancements.

The term “Generalized Euler Decomposition Formula” encompasses a spectrum of results that extend the archetypal Euler identities across algebra, geometry, topology, representation theory, and special function theory. Key instances include the Euler–Poincaré formula for convex polytopes, decomposition identites in shuffle and stuffle algebras for polylogarithms and multiple zeta values, matrix decompositions for Euler polynomial matrices, representation-theoretic decompositions in Lie theory, q-analogs, and effective-action generalizations in continuum physics. Each instantiation yields a structural decomposition—a sum or product expansion—anchored in the combinatorics, algebraic, or geometric structure of the objects under study.

1. Generalized Euler–Poincaré Formula for Convex Polytopes

The Euler–Poincaré formula gives a universal alternating sum across the $i$-dimensional faces of a convex $d$-polytope $P \subset \mathbb{R}^d$,

$\sum_{i=0}^d (-1)^i f_i = 1,$

where $f_i$ is the number of $i$-faces ($f_0$: vertices, $f_1$: edges, ..., $f_{d-1}$: facets, $f_d = 1$ for the polytope itself). This alternating sum is combinatorially interpreted as an inclusion–exclusion count. Topologically, it evaluates to the Euler characteristic $\chi$ of the $(d-1)$-sphere, since the boundary of a convex polytope is homeomorphic to $S^{d-1}$, and $\chi(S^{d-1}) = 1 + (-1)^{d-1}$.

A concise inductive proof (using projection and double-counting rather than shellability) yields the result for any $d \geq 1$ (Hliněný, 2016). The formula is fundamental in combinatorial topology, providing the basis for deeper relations among the $f$-vector quantities, such as the Dehn–Sommerville equations, and it underpins the algebraic Euler–Poincaré relation in homology. Computationally, it serves as a consistency check for face-number lists in convex polytopes and cell complexes.

2. Decomposition Formulas for Multiple Polylogarithms and Multiple Zeta Values

The Euler decomposition formula for products of Riemann zeta values,

$$\zeta(r)\zeta(s) = \sum_{u+v=r+s,~u,v\geq 1} \left[ \binom{u-1}{r-1} + \binom{v-1}{s-1} \right] \zeta(u,v)$$

admits a highly nontrivial generalization via shuffle and stuffle algebra structures. For multiple polylogarithm values (MPVs), an explicit generalized Euler formula expresses the product of two MPVs as a $\mathbb{Z}$-linear combination of MPVs of higher depth.

Let $G$ be a (multiplicative) abelian group and $s, t$ multi-indices, $z, w$ root tuples. The main formula involves a sum over shuffles $J_{k,\ell}$ and compositions of the weights, with binomial coefficients reflecting shuffle combinatorics: $$\operatorname{Li}_s(z)\operatorname{Li}_t(w) = \sum_{(\varphi, \psi)\in J_{k,\ell}} \sum_{u_1+\dots+u_{k+\ell}=|s|+|t|} \left( \prod_{i=1}^{k+\ell} \binom{u_i-1}{h_{(\varphi,\psi)}(i)-1} \right) \operatorname{Li}_{u_1,\ldots,u_{k+\ell}}(\xi_1,\ldots,\xi_{k+\ell})$$ where $h_{(\varphi, \psi)}(i)$ selects from $s$ or $t$ as dictated by the shuffle structure (Guo et al., 2008). This generalizes the classical depth-1 decomposition to arbitrary depth, foundational for the structure theory of multiple zeta values (MZVs) and polylogarithms.

3. Matrix and Polynomial Decomposition: Generalized Euler Matrices

Generalized Euler polynomials $E_n^{(\alpha)}(x)$ admit an addition (decomposition) law,

$$E_n^{(\alpha+\beta)}(x+y) = \sum_{k=0}^n \binom{n}{k} E_k^{(\alpha)}(x) E_{n-k}^{(\beta)}(y).$$

This induces a matrix factorization for the $(n+1)\times(n+1)$ Euler matrices $E^{(\alpha)}(x)$. Specifically,

$$E^{(\alpha+\beta)}(x+y) = E^{(\alpha)}(x) E^{(\beta)}(y) = E^{(\beta)}(x) E^{(\alpha)}(y),$$

and, for any $k$,

$$\left(E^{(\alpha)}(x)\right)^k = E^{(k\alpha)}(kx).$$

Euler polynomial matrices further decompose via the generalized Pascal matrix $P[x]$, Stirling matrices (of the first and second kind), Vandermonde matrices, and also through structured matrices built from Fibonacci and Lucas numbers. These decompositions yield a unified algebraic structure underlying identities across combinatorics and special functions (Quintana et al., 2018).

4. Generalized Decomposition in Representation Theory: Lie Algebra Tensor Powers

In the representation theory of the Lie algebra $A_n = \mathfrak{sl}_{n+1}(\mathbb{C})$, tensor powers of the adjoint representation exhibit a decomposition governed by Euler's difference table. Specifically, in the stable range $2k \leq n+1$,

$$X^{\otimes k} \cong \bigoplus_{j=0}^k c_j^{(k)} Y_j,$$

where $c_j^{(k)} = \binom{k}{j} d_{k-j}^{(j)}$ and $d_m^{(j)}$ are higher derangement numbers arising from Euler's difference table recurrence. This result draws a direct parallel between combinatorial decompositions in permutation enumeration and the decomposition of representation tensor products, highlighting the combinatorial underpinning of representation-theoretic structure (Perelomov, 2019).

5. q- and Interpolated Generalizations: qMZVs and t-shuffle Identities

q-analogs of Euler's decomposition, such as the formula for $q$-multiple zeta values (qMZVs), provide identities in noncommutative $q$-shuffle and stuffle algebras: $\mathfrak{z}_q(a)\mathfrak{z}_q(b) = \cdots~\text{(sum over double-depth qMZVs and %%%%34%%%% terms)}$ with precise coefficients as in (Medina et al., 2013).

Further, the theory of interpolated multiple zeta values (IMZVs) introduces a $t$-parameter interpolation between the classical and star MZVs, and the general t-shuffle product leads to a generalized Euler decomposition formula for IMZVs. This formula depends on the $t$-shuffle of words over $\{x, y\}$ and captures both classical and star versions as special cases. The structure involves intricate multinomial sums and reduction to classical shuffle formulas when $t=0$ (Sarkar et al., 31 Jan 2026).

6. Analytic and Algebraic Generalizations: Complex-Type Numbers and Exponential Decomposition

Generalizing the classical Euler identity to arbitrary “complex-type” imaginaries $h$ satisfying $h^2 = a + b h$, the exponential admits the decomposition: $e^{\theta h} = C(\theta) + h S(\theta)$ with closed-form expressions for $C, S$ in terms of the parameters $a, b$. Differential equations and addition formulas for $C, S$ generalize trigonometric identities. The geometric interpretation ties these decompositions to conic sections (ellipse, parabola, hyperbola) in the $(x, y)$-plane, determined by the discriminant of the quadratic satisfied by $h$ (Babusci et al., 2011).

7. Physical Generalizations: Fluid Dynamics and Effective Actions

In continuum physics, the generalized Euler equation derived from effective field theory for perfect fluids incorporates a geometric correction: $$\epsilon + p = s T + \mu q + y \frac{\partial p}{\partial y}$$ where $y$ encodes a geometric scale (e.g., curvature radius) of the background. This correction is purely geometric and vanishes in the flat-space (thermodynamic) limit. The same structure gives a local generalization of the Smarr formula for AdS black holes, demonstrating the universality of Euler-type decomposition in geometric thermodynamics (Mancilla, 2024).

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Across these domains, the Generalized Euler Decomposition Formula operates as a unifying principle: dissecting algebraic, combinatorial, and geometric structures into canonical summations, mappings, or tensor product decompositions. Its manifestations are central in polytope theory, shuffle and stuffle algebras, enumerative combinatorics, special function theory, representation theory, $q$-analysis, and continuum mechanics.