ExpMul Operator: Fast Exponential Methods

• ExpMul operator is a family of exponential-multiplicative constructs that replace infinite series with concrete, efficient algorithmic alternatives across diverse mathematical domains.
• It enables fast computation of truncated power series using Newton iteration and FFT, achieving optimal complexity in symbolic computation.
• In non-commutative settings like Lie groups and quantum dynamics, ExpMul offers closed-form, finite product decompositions that enhance numerical stability and analytical clarity.

The ExpMul operator is an umbrella term for a family of exponential-multiplicative constructs and algorithms arising in diverse areas: fast computation of power series exponentials, finite product formulas for non-commutative Lie group exponentials, resolvent-based integral operators in quantum dynamics, non-commutative *-exponentials for slice-regular functions, and exponential operators on arithmetic functions. Across these domains, ExpMul encapsulates methodologies that replace infinite expansions or non-constructive definitions with concrete, stable, and often optimally efficient algorithmic or structural alternatives.

1. Operator Definitions and Core Contexts

The specific realization of the ExpMul operator is context-dependent:

• Truncated Exponential of Power Series: Given $f(x)$ in a ring $K[[x]]$, $\operatorname{ExpMul}_n(f)$ is defined as $\exp(f(x)) \bmod x^n$, representing the truncated exponential series modulo $x^n$, i.e., $\sum_{k=0}^{n-1} \frac{f(x)^k}{k!}$ within $K[x]/(x^n)$ (Bostan et al., 2013, Sergeev, 2012).
• Lie Algebraic Exponentials: For non-commuting Lie algebra generators $X, Y, Z$ with commutators $[X,Y]=\kappa Z$, ExpMul refers to the canonical reduction $e^{aX+bY} = e^{pZ}e^{qX}e^{-pZ}$, with explicit $p$ and $q$, bypassing the classical Campbell–Baker–Hausdorff–Dynkin (CBHD) series in favor of a finite product (Kim et al., 28 Jan 2024).
• Quantum Integral Superoperator: In the quantum Liouvillian formalism, ExpMul is an integral operator representing the Laplace-weighted time integral of an observable, given by the resolvent $(\lambda I \pm \mathcal{L})^{-1}$ applied to an operator $A$ (Sturniolo, 2017).
• Slice-regular Quaternion Exponential: For slice-regular quaternionic functions, the ExpMul or $*$-exponential operator generalizes the exponential via the non-commutative $*$-product, yielding $\exp_*(f) = \sum_{n=0}^{\infty} \frac{f^{*n}}{n!}$ with rich trigonometric structure and non-trivial commutation properties (Altavilla et al., 2018).
• Arithmetic Function Operator: ExpMul is also instantiated as the exponential operator $E$ acting on arithmetic functions, mapping $f$ to $Ef$ via $(Ef)(p^a) = f(a)$ for all prime powers $p^a$, and extended multiplicatively (Lelechenko, 2013).

2. Fast Algorithms for Series Exponential (Newton Iteration and FFT)

The computation of truncated exponentials $\exp(f(x)) \bmod x^n$ is of central computational importance in symbolic algebra. Modern algorithmic strategies use Newton iteration combined with FFT-based polynomial multiplication, achieving soft-$O(M(n))$ runtime, where $M(n)$ is the multiplication complexity for degree $<n$ polynomials. The doubling approach progresses through block sizes $m=1,2,4,\dots,n$ (Bostan et al., 2013, Sergeev, 2012):

• At each stage, both $f = \exp(h) \bmod x^m$ and its inverse $g = 1/f \bmod x^{m/2}$ are maintained.
• The main iteration consists of a Newton step translated into four polynomial convolutions per doubling.
• The overall multiplication count per recursion is tightly bounded, leading to a proven constant factor (e.g., $(23/12+o(1))M(n)$ for exponentiation, $(27/8+o(1))M(n)$ for constant powers) (Sergeev, 2012).
• Asymptotic complexities achieve just $2.75$ times that of a single product in the FFT model.

This framework universally underpins fast computer algebra systems' approach to ExpMul-type computations.

3. Closed-Form Alternatives to CBHD for Non-Commutative Exponentials

In Lie group settings, particularly $\mathfrak{su}(2)$ and similar cyclically-closed algebras, ExpMul constructs a finite product representation for exponentials of the form $e^{aX+bY}$ or $e^{aX+bY+cZ}$. The essential results are (Kim et al., 28 Jan 2024):

• For $[X,Y]=\kappa Z$, $e^{aX+bY}$ has the decomposition:

$e^{aX + bY} = e^{pZ} e^{qX} e^{-pZ}$

with $p = \frac{1}{\kappa}\tan^{-1}(b/a)$, $q = \sqrt{a^2 + b^2}$.

• With three cyclic generators, $e^{aX+bY+cZ}$ is reduced to a finite product of five exponentials by two successive ExpMul operations:

$$e^{aX+bY+cZ} = e^{p_1Z}e^{q_1Y}e^{rX}e^{-q_1Y}e^{-p_1Z}$$

All required angles and coefficients are computed in closed form. This approach eliminates the need for the infinite CBHD series and yields numerically stable, analytically explicit propagators.

• These results are foundational for time-evolution operators in quantum spin systems (notably NMR, quantum computation) when the generators close in three dimensions.

4. Operator ExpMul in Quantum and Non-Commutative Function Theory

In quantum dynamics, ExpMul as the resolvent operator captures the Laplace transform of operator-valued expectation values (Sturniolo, 2017):

• For a Hamiltonian $H$ and observable $A$, the Laplace-weighted Heisenberg expectation integral is $$E(\lambda) = \langle\psi(0)|\mathcal{E}(\lambda)|\psi(0)\rangle$$ with $$\mathcal{E}(\lambda) = (\lambda I + \mathcal{L})^{-1}A$$.
• In the energy basis, this reduces to $$[\mathcal{E}(\lambda)]_{ij} = A_{ij}/[\lambda - i(\epsilon_i - \epsilon_j)/\hbar]$$.
• The method requires $O(N^3)$ (for matrix diagonalization and back-transformation), dramatically more efficient than direct quadrature if multiple damping rates or frequencies must be resolved.

Similarly, in quaternionic function theory, the $*$-exponential ExpMul is constructed via the power series in the non-commutative $*$-product of slice-regular functions (Altavilla et al., 2018):

• Explicit sine-cosine decompositions generalize the Euler formula to the non-commutative setting.
• A full classification of slice-preserving and $\mathbb{C}_J$-preserving exponential functions is achieved, and exceptional cases of multiplicativity (even in the absence of commutativity) are established.
• $\exp_*(f)$ is always invertible and never vanishing, with explicit trigonometric forms given suitable square roots of the symmetrized square.

5. ExpMul in Arithmetic Function Theory

The exponential operator $E$ on arithmetic functions provides a multiplicativity-preserving mechanism for redefining or iterating divisor functions and similar constructs (Lelechenko, 2013):

• $E$ maps $f$ to $Ef$, where $(Ef)(p^a)=f(a)$ and $Ef$ is extended multiplicatively.
• The $m$-fold iterate $E^m$ satisfies a nested-divisor formula:

$$(E^mf)(p^a) = \sum_{d_1\mid a} \sum_{d_2\mid d_1} \cdots \sum_{d_{m-1}\mid d_{m-2}} f(d_{m-1})$$

• For $f=\tau_k$, the $k$-fold divisor function, the generalizations $E^m\tau_k$ yield structured summatory asymptotics with explicit error exponents and facilitate sharp refinements for Dirichlet divisor problems.
• For large $m$, the action of $E^m$ on bounded functions eventually becomes identically $1$ on any initial segment of $\mathbb{N}$; for arbitrary $f$, $E^mf(n)\ll n^\epsilon$ for large $m$.

The exponential divisor construction thereby unifies a range of multiplicative arithmetic phenomena and underpins advanced results in divisor-sum estimates and related analytic number theoretic topics.

6. Applications and Computational Advantages

The ExpMul operator enables efficiency, tractability, and mathematical clarity across several disciplines:

• Symbolic Computation: Optimal algorithms for exponentiating power series or raising to constant powers, foundational for symbolic manipulation systems (Bostan et al., 2013, Sergeev, 2012).
• Quantum Control and Spectroscopy: Analytical decomposition of propagators in spin systems and fast resolvent computation for time-integrals of quantum observables (Kim et al., 28 Jan 2024, Sturniolo, 2017).
• Non-Commutative Function Theory: Closed-form generalizations of exponential functions to quaternions with a slice-regular structure, supporting further developments in quaternionic and Clifford analysis (Altavilla et al., 2018).
• Number Theory: New summation and estimation results for generalized divisor and totient functions via exponential operators on arithmetic functions (Lelechenko, 2013).

A consistent theme is the replacement of infinite series expansions, slow quadrature, or non-constructive definitions with constructive, finite, or algorithmically optimal alternatives.

7. Summary Table: Main ExpMul Operator Instantiations

Domain	ExpMul Definition	Notable Attributes and Results
Power series algebra	$\exp(f(x)) \bmod x^n$	$O(M(n))$ Newton/FFT algorithms, constant-factor optimal
Lie groups, $\mathfrak{su}(2)$	$e^{aX+bY} = e^{pZ}e^{qX}e^{-pZ}$	Closed-form, finite product, su(2)/so(3) closure
Quantum Liouvillian formalism	$(\lambda I + \mathcal{L})^{-1}A$	Resolvent formula, direct Laplace transform computation
Slice-regular quaternionic	$\exp_*(f) = \sum \frac{f^{*n}}{n!}$	Non-commutative $*$-exponential, trigonometric decompositions
Arithmetic functions	$(Ef)(p^a) = f(a)$, extend multiplicatively	Iterated divisor/totient structures, new asymptotics

The ExpMul conceptual framework thus spans and integrates computational algebra, non-commutative analysis, quantum theory, and analytic number theory through context-adapted exponential multiplicative machinery, enabling both theoretical and algorithmic advances (Bostan et al., 2013, Sergeev, 2012, Kim et al., 28 Jan 2024, Sturniolo, 2017, Altavilla et al., 2018, Lelechenko, 2013).