Holonomic Power Series

• Holonomic power series are formal power series whose coefficients satisfy linear recurrences with polynomial coefficients, defining them as D-finite functions.
• They are closed under addition, multiplication, and algebraic substitution, enabling effective algorithmic manipulation through methods like creative telescoping and Gröbner bases.
• Their applications span combinatorics, special functions, and formal language theory, linking analytic, algebraic, and computational methods in diverse mathematical settings.

A holonomic power series—also called a D-finite power series—is a formal power series over a field of characteristic zero whose coefficients satisfy a linear recurrence with polynomial coefficients, or equivalently, whose sum as a function satisfies a linear differential equation with polynomial coefficients. The concept, central to algebraic combinatorics, symbolic computation, and the theory of D-modules, is tightly linked to algorithmic properties, closure under algebraic operations, and concrete applications in combinatorics and mathematical physics. Holonomic functions are precisely those representable by finitely-generated left modules over the corresponding Weyl algebra of differential operators, and their generating functions encode a broad spectrum of combinatorial sequences and special functions (Bostan et al., 10 Dec 2025, Sattelberger et al., 2019, Wang et al., 2022).

1. Formal Definitions and Characterizations

For a field $K$ of characteristic zero (typically $\mathbb{Q}$ or $\mathbb{C}$), the univariate formal power series

$y(x) = \sum_{n=0}^\infty a_n x^n \in K[[x]]$

is called holonomic (or D-finite) if there exists a nonzero linear differential operator

$$L = p_r(x) D^r + \dots + p_0(x), \quad p_j(x) \in K[x],\, p_r \not\equiv 0,\ D = \frac{d}{dx}$$

such that $L \cdot y(x) = 0$. Equivalently, $(a_n)_{n\geq 0}$ satisfies a finite-order linear recurrence with polynomial coefficients: $$\sum_{j=0}^r p_j(n) a_{n+j} = 0, \quad p_j(n) \in K[n],\ p_r \not\equiv 0$$ This equivalence arises by applying $L$ term-wise to the series, leveraging $D^i x^n = n(n-1)\cdots(n-i+1)x^{n-i}$.

In several variables, for $$F(x_1, \ldots, x_n) = \sum_{\alpha \in \mathbb{N}^n} f_\alpha x_1^{\alpha_1} \cdots x_n^{\alpha_n} \in K[[x_1, \ldots, x_n]]$$, holonomicity means that for each $i$, there exists a nontrivial linear differential operator in $x_i$ (with polynomial coefficients in the $x_j$), annihilating $F$, and the $K(x_1, ..., x_n)$-vector space spanned by all mixed partials of $F$ has finite dimension (Sattelberger et al., 2019, Bostan et al., 10 Dec 2025).

A sequence or series is called P-recursive when its coefficients satisfy such a linear recurrence; the minimal order $J$ is the order of the holonomic sequence (Wang et al., 2022).

2. Structural and Closure Properties

The class of holonomic power series is robust under key algebraic and analytic operations:

• Addition and multiplication: Closed under finite sums and products; if $F$ and $G$ are holonomic, so are $F+G$ and $F\cdot G$.
• Hadamard product: The coefficientwise product, $(F \odot G)(x) = \sum_n f_n g_n x^n$, is holonomic.
• Cauchy (convolution) product: The ordinary product of series is holonomic.
• Algebraic substitution: If $F(x)$ is holonomic and $\alpha(x)$ is algebraic, then $F(\alpha(x))$ is holonomic.
• Specialization/diagonalization: Extracting the diagonal or specializing parameters preserves holonomicity.
• Differentiation and shift: Termwise differentiation and difference preserve holonomicity.
• Hypergeometric-type sequences: Sequences where $a_{n+m}/a_n$ is rational (for some $m$) admit an $m$-fold symmetric structure in their recurrence, which falls within the holonomic framework (Tabuguia et al., 2021, Wang et al., 2022).

All algebraic (and hence rational) series are holonomic. Combinatorial sequences such as binomial coefficients, Domb numbers, Franel numbers, and Delannoy numbers are holonomic, as are classical special functions like the exponential, binomial, and Bessel functions (Sattelberger et al., 2019, Wang et al., 2022).

3. Algorithmic and Symbolic Computation Aspects

Holonomic power series—owing to their well-structured recurrence/differential relations—lend themselves to effective algorithmic manipulation:

• Gröbner bases in Weyl algebra: The set of annihilating operators for $f$ (the $D$-ideal) allows elimination and manipulation using Buchberger's algorithm adapted for the noncommutative setting (Sattelberger et al., 2019).
• Creative telescoping: Zeilberger's and Chyzak's algorithms systematically derive recurrences/differential equations for sums and integrals involving holonomic functions, crucial for evaluating definite multi-sums or integrals (Wang et al., 2022, Sattelberger et al., 2019).
• Polynomial reduction: For holonomic recurrences, the polynomial-reduction algorithm determines when a weighted term $q(n)F(n)$ can be written as a telescoping sum, essential for deriving closed-form expressions for sums and congruences—extending the methodology from hypergeometric to general holonomic sequences (Wang et al., 2022).
• Hypergeometric–type series and algorithms: The $mfoldHyper$ algorithm generalizes van Hoeij's solver for hypergeometric solutions of holonomic recurrences to compute all $m$-fold hypergeometric term solutions, structuring solutions and supporting representation and automated computation of a broad class of power series (Tabuguia et al., 2021).

The computational complexity of these algorithms is generally polynomial in the degree of recurrences and weights for single-sum identities, but may become more demanding (e.g., doubly-exponential) for multi-variate or nested sums due to Gröbner basis computations (Wang et al., 2022, Bostan et al., 10 Dec 2025).

4. Applications and Examples

Holonomic series are foundational in enumerative combinatorics, analytic number theory, symbolic summation, and mathematical physics:

• Generating functions of automata: The class of languages recognized by weakly-unambiguous Parikh automata has holonomic generating series, and the univariate (length-counting) series is always holonomic (Bostan et al., 10 Dec 2025).
• Special functions and identities: The exponential, Bessel, and binomial series are realized holonomically, each satisfying specific linear ODEs/recurrences (Sattelberger et al., 2019).
• Summation identities and $\pi$-series: The polynomial-reduction framework allows for the derivation of closed-form and $\pi$-related series for Domb numbers and Franel numbers. For instance,

$$\sum_{n=0}^\infty \frac{n^2(n-1)(9n+1)}{(-32)^n}\,\mathrm{Domb}(n) = \frac{4\pi}{3}$$

with similar parameterized constructions for Franel numbers (Wang et al., 2022).

• Multi-summation and creative telescoping: Nested applications of reduction and telescoping yield explicit expressions for double or higher sums involving holonomic sequences.

5. Holonomic Series in Formal Language Theory

The interplay between holonomic series and formal languages is exemplified by Parikh automata theory:

• Automata-theoretic characterization: Weakly-unambiguous Parikh automata (PA) correspond precisely to unambiguous two-way reversal-bounded counter machines. Their multivariate Parikh generating series are holonomic (Bostan et al., 10 Dec 2025).
• Non-equivalence of converses: There exist languages whose counting (generating) series are holonomic but which are inherently weakly-ambiguous as PA languages; holonomicity of the generating series does not guarantee recognizability by weakly-unambiguous PA. This is witnessed by classes of deterministic context-free languages constructed to enforce ambiguity at every level.
• Algorithmic inclusion: The decidability of inclusion for weakly-unambiguous PA is enabled by holonomic properties: inclusion reduces to finitely many checks on the vanishing of initial coefficients, due to a bound given by the annihilating ODE’s parameters. The complexity is doubly-exponential in the dimension and size parameters of the automata (Bostan et al., 10 Dec 2025).

6. Limitations and Contemporary Research Directions

Despite their closure robustness, holonomic series do not encompass all analytic or combinatorial functions of interest:

• Non-holonomic series: Important functions such as $\tan z$, $\sec z$, $\log(1+\sin z)$, or $z/(e^z-1)$ are not D-finite; their coefficients obey non-linear recurrence relations, such as those derived from quadratic differential equations and convolution sums (Tabuguia et al., 2021).
• Algorithmic frontiers: Extending symbolic summation and reduction to $q$-holonomic (basic hypergeometric) sequences remains an open field, as does improving the complexity of multivariate creative telescoping (Wang et al., 2022).
• D-module generalizations: The theory of holonomicity extends to D-modules, with computational frameworks centered on annihilators, characteristic varieties, and algorithmic solutions using Gröbner bases—influencing applications ranging from algebraic geometry to mathematical statistics (Sattelberger et al., 2019).

7. Representative Examples

Name	Recurrence/ODE	Initial Terms/Form
Exponential ($e^x$)	$D y - y = 0$	$a_{n+1} = a_n/(n+1)$, $a_0=1$
Binomial $(1-x)^{-\alpha}$	$(1-x)D y - \alpha y=0$	$a_{n+1} = (n+\alpha)a_n/(n+1)$
Bessel $J_\nu(x)$	$x^2 D^2 y + x D y + (x^2 - \nu^2)y=0$	Rec. order 2; see details
Domb numbers	3-term rec.; see (Wang et al., 2022)	$$\mathrm{Domb}(n) = \sum \binom{n}{k}^2 \binom{2k}{k}$$
Franel numbers	3-term rec.; see (Wang et al., 2022)	$f(n) = \sum \binom{n}{k}^3$
Delannoy numbers	3-term rec.; see (Wang et al., 2022)	$D(n) = \sum \binom{n}{k} \binom{n+k}{k}$

This highlights the breadth of holonomic series, their algorithmic amenability, and their central role in the computational theory of special and combinatorial functions (Sattelberger et al., 2019, Wang et al., 2022, Bostan et al., 10 Dec 2025).