Power Series Approach

• Power series methods are representations of functions as infinite sums with coefficients derived from problem structure, widely used in analysis and differential equations.
• They enable efficient algorithmic operations such as composition, inversion, and summation, significantly optimizing computer algebra and model reduction tasks.
• Applications extend to solving ODEs/PDEs, system realization in control theory, and even number theory, illustrating both theoretical depth and practical utility.

A power series approach in mathematics refers to the systematic use of formal or convergent expansions of functions (often analytic functions, operators, or solutions of equations) as infinite sums of powers of variables, typically with coefficients determined by the structure of the underlying problem. These approaches play a central role in analysis, differential equations, algebra, system theory, symbolic computation, and model reduction.

1. Fundamental Concepts of Power Series Methods

A power series in one variable is an infinite sum of the form

$f(x) = \sum_{n=0}^\infty a_n x^n,$

where $a_n$ are (often field-valued) coefficients. In several variables, the expansion generalizes to

$$f(x_1,\dots,x_d) = \sum_{n_1,\dots,n_d\ge 0} a_{n_1,\dots,n_d} x_1^{n_1} \cdots x_d^{n_d}.$$

A power series is said to converge within its radius of convergence, but many techniques operate formally, independent of convergence.

Power series methods encompass:

• Constructive algorithms for function composition, inversion, and manipulation of (possibly multivariate) power or Laurent series;
• Analytic or semi-analytic representations of solutions to ODEs, PDEs, or fractional differential equations via truncations or resummations;
• Algebraic encoding of dynamical systems and network models, especially via non-commutative formal series and generating functions;
• Symbolic and algorithmic frameworks suitable for model reduction, system realization, and computational applications.

2. Symbolic and Algorithmic Frameworks

Algorithmic manipulation of power series is foundational in computer algebra and symbolic computation. Recent advances include:

Near-linear-time power series composition:

The computation of the truncated composition $f(g(x)) \bmod x^n$ for two given series $f(x)$ and $g(x)$ with $g(0)=0$ is a classical symbolic computation problem. The 2024 algorithm by Bostan et al. yields a cost of $O(\mathrm{Mul}(n)\log m + \mathrm{Mul}(m))$, where $\mathrm{Mul}(n)$ is the cost of polynomial multiplication of degree $n$ over a base ring. This is achieved via an adaptation of the Graeffe iteration, yielding the first purely algebraic near-linear algorithm for truncated power series composition (Kinoshita et al., 2024):

Algorithm	Complexity
Brent–Kung (1978)	$O(\sqrt{n\log n}\cdot \mathrm{Mul}(n))$
Kedlaya–Umans (2008)	$O((n\log q)^{1+o(1)})$ (bit)
N-SSV (2023)	$\tilde{O}(n^{1.43})$ (algebraic)
Bostan et al. (2024)	$\tilde{O}(n)$ (algebraic)

Key subroutines involve polynomial reciprocal computation, Kronecker reduction for bivariate multiplication, and the transposition principle for linear programs.

Inversion of formal power series:

Algorithmic inversion of a formal power series $f(x) = x + H(x)$ ($H$ order $\geq 2$) up to degree $D$ proceeds via I-adic iteration and the construction $P_{k+1}(x) = P_k(f(x)) - P_k(x)$. This converges I-adically, giving $g(x) = f^{-1}(x) \bmod x^{D+1}$ with complexity $O(D^2\log D)$ for fast composition (Adamus, 5 Mar 2025).

Summation using recurrence relations:

Given a power series whose coefficients satisfy a linear recurrence with polynomial coefficients, one can algorithmically derive and solve the associated linear differential equation for the generating function,

$$Q_0(x)F(x) + Q_1(x)F'(x) + \cdots + Q_M(x)F^{(M)}(x) = R(x),$$

enabling closed-form summation of a wide class of series (Talvila, 15 Jan 2026).

Power series of functions via derivative matching:

Approximating $f(x)$ using a series $\mathcal{A}^f(x) = \sum_{n=0}^\infty a_n [g(x)]^n$, with coefficients computed via generalized Faà di Bruno formula and Bell polynomials (second kind), produces expansions that may converge more rapidly or on a larger domain than ordinary Taylor series, especially for suitable choices of $g$ (Liptaj, 2022).

3. Power Series in the Analysis of Differential Equations

Ordinary and Partial Differential Equations

Power series approaches are widely employed in the analysis of ODEs and PDEs, especially for constructing local solutions about regular points:

• Nonlinear PDEs:

The power series solution for a nonlinear PDE involves: (1) expressing the dependent variable as a multivariate Taylor series, (2) substituting into the PDE, (3) expanding nonlinear products via Cauchy product, and (4) matching coefficients of monomials to derive recursion relations for the series coefficients. Analyticity of the equation and initial/boundary data ensures local convergence, with the radius estimable via the Cauchy–Hadamard formula (Lopez-Sandoval et al., 2012).

• Inverse analysis in chemical kinetics:

The power series method is used to solve inverse problems for kinetic ODEs, simultaneously determining both the analytic time-dependent concentration profiles and unknown kinetic parameters by matching recurrences and experimental data. This formalism transforms the inverse problem into a nonlinear algebraic system for coefficients and parameters (Lopez-Sandoval et al., 2012).

• Fractional differential equations:

The Asad Correctional Power Series Method (ACPS) constructs solutions to Caputo-type fractional ODEs by enforcing vanishing of the defect (difference between the Caputo derivative of the series and the RHS) and all its fractional derivatives at the expansion point, leading to recursive determination of series coefficients. This approach encodes nonlocal memory effects unique to fractional calculus and demonstrates rapid convergence and high accuracy even compared to dense numerical integrators (Freihet et al., 29 Sep 2025).

• Power-series-based simulation in applied systems:

Semi-analytical power series expansions (SAS) in time-domain simulation of power systems yield closed-form, parameterized, and efficiently evaluable approximations to nonlinear dynamics, with rigorously controlled error bounds and adaptive time stepping for large-scale DAE networks (Wang et al., 2018).

Asymptotic Approximants and Divergent Series

In cases where the natural power series diverges, or only a truncated expansion is available, power-series-based asymptotic approximants bridge local expansions with known asymptotic behavior at infinity (or another singularity). The approximant is constructed to match both the truncated series and the outer asymptotic form, enabling accurate global solutions and prediction of unknown parameters or singularities (Barlow et al., 2017).

4. Power Series in Systems, Control, and Realization Theory

Formal power series in system realization:

Non-commutative formal power series over a finite alphabet (such as Chen–Fliess series) provide a unifying representation for analytic input–output maps of (nonlinear or hybrid) dynamical systems. Partial realization theory builds minimal reachability and observability representations directly from formal series coefficients using truncated Hankel matrices, facilitating identification and reduction of (switched or hybrid) systems (Petreczky et al., 2010).

Moment-matching-based model reduction:

For high-order MIMO polynomial nonlinear systems, power-series decomposition solves the center manifold PDE by reducing it to a recursive sequence of Sylvester equations for the coefficients. The approach yields reduced-order models with explicit moment-matching guarantees up to a prescribed degree, and reveals algebraic limits on achievable model reduction orders for MIMO systems depending on the input/output channel structure (Huang et al., 19 Aug 2025).

Multiplicative dynamic and static feedback:

Non-commutative formal power series methods provide explicit group-theoretic formulas for the closed-loop generating series in feedback interconnections. The composition uses transformation-group actions in the algebra of formal series and is structurally controlled by dual Hopf and pre-Lie algebra frameworks. This algebraic formalism underlies modern treatments of nonlinear feedback invariants and closed-loop realization (Venkatesh, 2022, Ebrahimi-Fard et al., 2023).

5. Applications to Number Theory and Beyond

Two-variable power series and the Riemann Hypothesis:

Power series techniques can encode profound arithmetic information—for example, via the two-variable $y$-Borel transform of the Riemann Xi function. Analyses of realness and simplicity of the zeros of such two-parameter series connect directly to the Riemann hypothesis, offering new reformulations and analogues of the de Bruijn–Newman constant (Brugidou, 2012).

6. Convergence, Acceleration, and Limitations

The success of power series approaches hinges on analyticity and the radius of convergence. For local PDE/ODE expansions, the domain of validity is bounded by singularities or analytic continuation barriers. For divergent series or problems with limited series data, acceleration methods such as Borel summation, Padé approximants, or matched asymptotics (as in the asymptotic-approximant method) extend the utility of the local series. In inverse or identification settings, care must be taken to ensure that auxiliary data or constraints are sufficient to render the resulting algebraic system well-posed (Barlow et al., 2017, Lopez-Sandoval et al., 2012).

Significant algorithmic advances have reduced the cost of power series operations, yet the curse of dimensionality remains acute for multivariate expansions, and local series may remain impractical for problems with non-analytic features such as shocks or discontinuities (Lopez-Sandoval et al., 2012).

7. Outlook and Extensions

Recent works indicate that the rational/Graeffe-based iteration paradigm may generalize to multivariate or non-commutative settings, suggesting avenues for near-linear complexity in more general modular composition problems and for symbolic manipulation in computational algebra. The power series formalism is also a backbone for algebraic analysis, hybrid system identification, and analytic model reduction in high-dimensional nonlinear systems, with ongoing developments in the abstract algebraic interpretation of system operations via Hopf, pre-Lie, and group-theoretic structures (Kinoshita et al., 2024, Ebrahimi-Fard et al., 2023).