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Holonomic function

A holonomic function is a function of one or more variables that satisfies a linear homogeneous (or system thereof) with coefficients, meaning its values and derivatives are linearly related through such equations. This class encompasses solutions to systems in the sense of theory, where the annihilator ideal in the Weyl algebra is , implying finite-dimensional spaces over the constants. Holonomic functions arise prominently in the study of linear equations with rational coefficients, as their solutions remain holonomic after clearing denominators to obtain coefficients. Key properties of include closure under fundamental operations such as , , , indefinite , and with rational functions, making them a stable class for algebraic manipulations. For instance, the spanned by a and its derivatives is finite-dimensional, and the function admits asymptotic expansions at regular singular points. Examples abound in , including the e^x, which satisfies f' - f = 0; the J_\nu(x), solving x^2 y'' + x y' + (x^2 - \nu^2) y = 0; and the {}_pF_q, which obeys a linear of order q+1. In the discrete setting, q- generalize this concept, satisfying linear recurrences with coefficients that are polynomials in q and q^n, and they share analogous closure properties under summation and product. Holonomic functions play a central role in symbolic computation and the automated proof of identities for , facilitated by algorithms like creative telescoping introduced by Zeilberger in the 1990s. Their theory underpins advances in cohomology and , with applications extending to quantum invariants, such as the colored Jones polynomial being q-holonomic. This framework also aids in evaluating definite integrals and sums involving parameters, as these yield holonomic systems of PDEs.

Fundamental Concepts

Definition for Functions

In , a holonomic function of one complex variable is a or f(z) that satisfies a linear homogeneous with coefficients, specifically of the form P_0(z) f^{(n)}(z) + P_1(z) f^{(n-1)}(z) + \cdots + P_n(z) f(z) = 0, where the P_i(z) are polynomials in z, P_0(z) is nonzero, and n denotes the order of the equation. This equation relates the function and its derivatives up to order n through polynomial dependencies, distinguishing holonomic functions from more general classes like arbitrary s. The n of such an is minimal if no linear homogeneous of lower with coefficients is satisfied by f(z). The minimal characterizes the function's "complexity" within this class, corresponding to the of the solution space spanned by a fundamental set of solutions near regular points. For instance, exponential functions like e^z satisfy a , making their minimal 1. The term "," originating from concepts in and referring to integrable constraints, was adapted by Doron Zeilberger in to describe these and systems in the context of proving identities via holonomic systems. The collection of all functions forms a over the field of rational functions \mathbb{C}(z), meaning that linear combinations with rational coefficients remain holonomic, with the minimal order of the combination at most the maximum of the individual orders.

Definition for Sequences

A sequence (a_n)_{n \geq 0} of numbers is if it satisfies a linear homogeneous with coefficients, specifically, there exist P_0(n), P_1(n), \dots, P_k(n) in \mathbb{C}, where k is a positive and P_0(n) is not the zero polynomial, such that for all sufficiently large n, P_0(n) a_{n+k} + P_1(n) a_{n+k-1} + \cdots + P_k(n) a_n = 0. The minimal such k is called the order of the recurrence (or the order of the sequence). sequences are also known as P-recursive sequences, emphasizing the polynomial nature of the coefficients in the defining recurrence. The ordinary generating function A(x) = \sum_{n=0}^\infty a_n x^n of a holonomic sequence is itself a holonomic function, meaning it satisfies a linear homogeneous differential equation with polynomial coefficients. Given the recurrence of order k and initial values a_0, a_1, \dots, a_{k-1}, the holonomic sequence is uniquely determined for all n \geq k.

Relation to Differential Equations and Recurrences

Holonomic functions are defined as the smooth solutions to systems of linear homogeneous partial equations with coefficients. This characterization establishes a direct correspondence between such functions and the s in the Weyl algebra generated by the operators that annihilate them. Specifically, for a function f of one variable, the Weyl algebra D = \mathbb{C}\langle \partial \rangle, where \partial = d/dx, acts on functions via the relation [\partial, x] = 1. The annihilator \mathrm{Ann}_D(f) = \{ P \in D \mid P \cdot f = 0 \} determines the D / \mathrm{Ann}_D(f), and f is if and only if this is , meaning its characteristic variety has equal to the of the (here, 1 for univariate functions). This framework, rooted in , provides the theoretical between functions and linear equations with coefficients, often studied through extensions of rings to incorporate non-commutative operators. In the discrete setting, holonomic satisfy linear homogeneous recurrence relations with polynomial coefficients in the independent variable. For a (a_n)_{n \geq 0}, this means there exist polynomials p_0(n), \dots, p_r(n) \in \mathbb{C}, not all zero, such that \sum_{k=0}^r p_k(n) a_{n+k} = 0 for all sufficiently large n. This equivalence mirrors the continuous case but uses shift operators in an algebra \mathbb{C}\langle S \rangle, where S a_n = a_{n+1}, instead of . The annihilator ideal in this algebra plays an analogous role, capturing the relations that the sequence obeys. Seminal work by Richard Stanley introduced this notion for P-recursive sequences, later termed holonomic, emphasizing their closure under shifts and rational operations. A between the continuous and realms is provided by s. The f(x) = \sum_{n=0}^\infty a_n x^n of a holonomic sequence (a_n) is itself a holonomic function, satisfying a with coefficients derived from the recurrence via operator transformations in the Ore algebra. Conversely, if f(x) is holonomic, applying the shift to the coefficients transforms the differential equation into a recurrence, establishing the equivalence. This connection enables algorithmic translations between differential equations and recurrences, facilitating proofs of identities and asymptotic analyses. A fundamental underscores this duality: a f(x) = \sum_{n=0}^\infty a_n x^n / n! (or ordinary ) is if and only if its coefficients (a_n) form a . This result follows from the fact that on f corresponds to shifts and multiplications by n on the coefficients, preserving the of the annihilator across the continuous-discrete boundary. Leonard Lipshitz's work on D-finite functions formalized this for , proving closure properties that reinforce the .

Properties and Characterizations

Closure Properties

Holonomic functions, also known as D-finite functions, exhibit remarkable stability under a variety of algebraic and operations. The class is closed under addition and : if f(x) and g(x) are , then so are af(x) + bg(x) for any constants a, b \in \mathbb{C}. Similarly, the product f(x) g(x) is , as the annihilating for the product can be derived from those of f and g using elimination techniques. These properties extend to and ; the f'(x) satisfies a obtained by applying the annihilator to the differentiated form, and the indefinite \int f(x) \, dx (or definite integral from 0 to x) is , with the operator adjusted via . For sequences, which satisfy linear recurrences with coefficients (P-finite sequences), holds under addition and by s. If (a_n) and (b_n) are , then (a_n + b_n) is , and multiplication by a r(n), which itself satisfies a recurrence, yields another sequence via the for recurrences. The class is also closed under shifts: if (a_n) is , then so is (a_{n+1}), as shifting preserves the recurrence structure. Further closures include the Hadamard product for sequences, where the termwise product (a_n b_n) is , mirroring the multiplication closure. For functions, the Hadamard product of representations is also holonomic under suitable convergence conditions. Composition is preserved under restrictions: if f(x) is holonomic and g(x) is algebraic with g(0) = 0, then f(g(x)) is holonomic, computable via resultant-based elimination of the intermediate variable. The class of holonomic functions and sequences is closed under taking limits of solutions to their defining equations, ensuring that or uniform limits of holonomic objects, when they satisfy the appropriate regularity, remain within the class. This under limits underscores the robustness of holonomic representations in asymptotic and computational contexts.

Uniqueness and Characterization Theorems

One key characterization of holonomic functions arises from the structure of the ideal in the Weyl algebra. Specifically, every holonomic function satisfies a linear of minimal order with coefficients, and this minimal-order operator is unique up to multiplication by a nonzero scalar in the . This uniqueness follows from the fact that the left ideal generated by the annihilators in the Weyl algebra admits a generator of minimal degree, providing a normal form for the satisfied by the function. A fundamental characterization theorem states that a smooth function f of several variables is holonomic if and only if the \mathbb{C}(x_1, \dots, x_n)- spanned by f and all its partial has finite dimension. This finite-dimensionality condition captures the algebraic constraints imposed by the holonomic system and distinguishes holonomic functions from more general analytic functions, whose derivative spans would be infinite-dimensional. For sequences, Zeilberger's theorem on creative telescoping provides a powerful characterization and construction tool. It asserts that if a bivariate F(n, k) is —meaning it satisfies a linear recurrence with coefficients jointly in n and k—then the definite sum S(n) = \sum_k F(n, k) (over a suitable finite range for k) is also in n. The proof involves algorithmically producing a telescoping relation that yields a recurrence for S(n) directly from the of F, enabling the certification of holonomicity for sums without explicit computation.

Examples

Holonomic Functions and Sequences

Holonomic functions and sequences are ubiquitous in mathematics, appearing in , , and their generating functions. These examples illustrate how linear equations or recurrences with coefficients capture a wide range of classical objects. A fundamental example is the e^z, which satisfies the first-order linear \frac{d}{dz} f(z) - f(z) = 0, with coefficients (of degree 0 in the leading term). This makes e^z of order 1. Bessel functions of the first kind, J_\nu(z), provide another classical instance. They satisfy Bessel's differential equation, z^2 \frac{d^2}{dz^2} y(z) + z \frac{d}{dz} y(z) + (z^2 - \nu^2) y(z) = 0, a second-order linear equation with polynomial coefficients in z, confirming their holonomicity. The generalized hypergeometric functions {}_p F_q \left( \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} ; z \right) are also holonomic, as they obey a linear differential equation of order q+1 with polynomial coefficients derived from their series definition. Turning to sequences, the n! is , satisfying the linear recurrence s(n+1) = (n+1) s(n), \quad s(0) = 1, where the coefficient is a in n. coefficients \binom{n}{k}, for fixed k, form a as polynomials in n of degree k, satisfying a linear recurrence of order k+1 with coefficients; this follows from identities like Pascal's recurrence \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}, which extends to a univariate form via creative telescoping. Generating functions for sequences, such as (1 - z)^{-k} = \sum_{n=0}^\infty \binom{n+k-1}{k-1} z^n, are as they coincide with the {}_1 F_0 (k ; ; z), satisfying an appropriate .

Non-Holonomic Functions and Sequences

A classic example of a non- function is e^{1/z}, which exhibits an at z = 0. Unlike functions, which satisfy linear s (ODEs) of finite order with coefficients and thus possess only regular singularities, e^{1/z} cannot be annihilated by any such ODE because its expansion around 0 involves infinitely many negative powers without a finite principal part, requiring either infinite order or non- coefficients for description. This irregular behavior at the prevents it from belonging to the class of functions. For sequences, the partition p(n), which enumerates the number of ways to write the positive n as a of positive integers disregarding , serves as a prominent non-holonomic example. The ordinary \sum_{n=0}^\infty p(n) z^n does not satisfy any linear with coefficients, as proven by showing that assuming such an equation leads to a with known modular properties of the . Consequently, the sequence p(n) fails to obey a linear with coefficients, highlighting its irregular growth pattern that defies the structured constraints of holonomic sequences. These non- cases are distinguished from ones partly by their asymptotic behaviors; for instance, while sequences typically admit controlled growth via recurrences, p(n) displays super-exponential asymptotics roughly on the order of \exp(c \sqrt{n}) for some constant c > 0, underscoring the absence of finite-order relations. Similarly, the rapid, direction-dependent growth of e^{1/z} near its further illustrates the to meet criteria.

Advanced Extensions

Multivariable Holonomic Functions

In the multivariable setting, a function f(x_1, \dots, x_m) of several variables is if it is annihilated by a left ideal in the Weyl D = \mathbb{C}[x_1, \dots, x_m]\langle \partial_1, \dots, \partial_m \rangle, where the partial derivatives \partial_i = \frac{\partial}{\partial x_i}. This means f satisfies a system of linear partial differential equations (PDEs) with polynomial coefficients, and the associated D-module has a characteristic of dimension m in \mathbb{C}^{2m}. Such functions arise naturally in , where the holonomicity ensures that the solution space is finite-dimensional over generic points, generalizing the finite-order condition from the single-variable case. A key tool for characterizing holonomicity in the multivariable context is the Bernstein-Sato polynomial, also known as the b-function, which is the b_f(s) \in \mathbb{C} of minimal satisfying a P \cdot f^{s+1} = b_f(s) \cdot f^s for some non-zero P \in D. The roots of b_f(s) are negative rational numbers, and their properties reflect the of the , distinguishing functions by ensuring the D-module's regularity and integrability along hypersurfaces. Unlike simpler cases, the multivariable b-function captures interactions across variables, aiding in the verification of holonomicity through computational algebraic methods. Prominent examples of multivariable holonomic functions are the A-hypergeometric functions introduced by Gel'fand, Kapranov, and Zelevinsky, which solve the GKZ system of PDEs defined by an integer matrix A \in \mathbb{Z}^{d \times n} and parameter vector \boldsymbol{\beta} \in \mathbb{C}^d. These functions generate holonomic ideals in the , with their solution spaces encoding combinatorial data from toric varieties and appearing in statistical models like the Fisher-Bingham distribution. The holonomic nature of GKZ systems follows from the finite rank of the associated , typically equal to the normalized volume of the toric . The multivariable framework connects to the one-variable case through restrictions: if f(x_1, \dots, x_m) is , then its restriction to a , such as setting x_{k+1} = \dots = x_m = c for constants c, yields a holonomic function in the remaining k variables. This preservation of holonomicity under non-characteristic restrictions allows reduction to lower-dimensional problems while maintaining the D-module's structural properties.

Connection to D-Finite Functions

In the algebraic framework, D-finite functions are defined as those that are annihilated by a non-zero linear with coefficients in the Weyl algebra, which consists of over the . This definition aligns precisely with the notion of functions in the single-variable case, where both terms describe functions satisfying linear equations with coefficients, ensuring a finite-dimensional solution space spanned by the function and its derivatives. For multivariable extensions, the concept generalizes to holonomic D-modules, which are left modules over the Weyl that possess finite length in the of D-modules; this finiteness implies a bounded with simple subquotients, capturing the of solutions to systems of partial equations. Such modules maintain the property through their characteristic varieties, which have minimal equal to that of the ambient space. Chyzak's theorem establishes that D-finiteness is preserved under definite and of D-finite functions, providing an algorithmic via creative telescoping to compute the annihilating differential operators for the resulting expressions. This preservation extends Zeilberger's methods from hypergeometric to general holonomic settings, enabling efficient symbolic computation of closure properties. Unlike global analytic functions, which emphasize convergence and holomorphy over entire domains, D-finite functions prioritize solutions to differential equations, focusing on algebraic and computational aspects without requiring .

Computational Methods

Algorithms for Computation and Verification

Algorithms for determining whether a function or is involve computing candidate annihilating s, such as linear recurrences or differential equations, and verifying if they reduce the object to zero. For s, Zeilberger's provides a method to find linear recurrences with coefficients from s, particularly for hypergeometric terms. The , introduced in the context of the systems approach, constructs a certificate that proves identities and derives recurrences by solving for undetermined coefficients in a derived from shift s. Specifically, for a defined by a S(n) = \sum_k F(n,k), where F is , the method applies creative telescoping to find a recurrence of the form P(\mathbf{N}, n) S(n) = 0, where \mathbf{N} denotes forward shifts and P is a in the shift and s. The steps include assuming an for the recurrence, setting up equations from the telescoping condition \mathcal{L}(n,k) F(n,k) = \mathbf{N}_k G(n,k) - G(n,k) for some \mathcal{L}, and solving the resulting over . This approach is efficient for low s, with complexity dominated by the solution of systems of size proportional to the assumed and . For continuous holonomic functions represented by power series, a linear algebra-based method, applied notably to the susceptibility, computes the minimal satisfied by the series. This method assumes a of order r with coefficients of degree at most d, expands the operator applied to the truncated power series y(x) = \sum_{i=0}^N a_i x^i, and sets the coefficients of the resulting series to zero, yielding a for the unknown coefficients. The system is overdetermined for sufficient N \gg r(d+1), and the minimal equation is obtained by finding the lowest r and d for which a non-trivial solution exists, solved via . The complexity is O((r(d+1))^3 + r^2 d N), in the series length N and parameters, allowing computation for series up to thousands of terms. Verification involves substituting the full series (or more terms) and confirming the result is identically zero within numerical precision or symbolically for rational coefficients. Creative telescoping extends these ideas to compute recurrences or s for definite sums and s of functions, enabling operations like indefinite or within the holonomic class. Originating from Zeilberger's work on hypergeometric sums, the general for holonomic functions, as extended by Chyzak, operates in the Weyl algebra by constructing an operator T that annihilates the sum or integral by eliminating the summation/ variable through telescoping relations. Key steps include parameterizing candidate telescopers as undetermined coefficient polynomials in differential and shift operators, substituting into the annihilation condition derived from the input holonomic system's operators, and solving the resulting using Gröbner bases or linear algebra to find the minimal-degree solution. For an input holonomic function f(z) satisfying a differential equation and an I(z) = \int f(z,t) \, dt, the outputs a differential equation for I(z), preserving holonomicity. The approach applies similarly to sums, with complexity in the number of variables due to ideal computations but practical for low-dimensional cases via heuristics for coefficient bounds. Verification of holonomicity, or whether a given annihilates a /, relies on ideal membership testing in the corresponding Ore . For a candidate recurrence or P, one computes a of the annihilator generated by known operators (if any) and reduces P with respect to the basis; if the remainder is zero, P annihilates the object. For or term s, an efficient numeric check substitutes the first M terms, where M exceeds the 's degree, and confirms the output coefficients vanish, with error bounded by machine precision. Symbolically, for rational inputs, exact arithmetic verifies the identity. The complexity of computation in the Weyl is double-exponential in the number of variables and operator degrees, limiting practical verification to small systems, though for univariate cases, it reduces to polynomial time via linear on Hankel-like matrices.

Software Implementations

The gfun package in facilitates the manipulation of sequences and functions defined by linear recurrences or equations with coefficients. Key features include guessing minimal recurrences or equations from a finite number of initial terms using the gfun[guess] command, computing expansions via gfun[series], and performing algebraic substitutions on functions with gfun[algebraicsubs]. These tools are particularly useful for rigorous computations in and , as detailed in the original implementation by Salvy and . Complementing gfun, Maple's DEtools package supports computations for holonomic differential equations through functions like DEtools[polysols] and DEtools[ratsols], which identify and rational solutions of linear ordinary differential equations (s) by solving associated systems in the Weyl algebra. The DEtools[FindODE] command determines the minimal-order homogeneous satisfied by a given , aiding verification of holonomicity. These capabilities enable efficient handling of solution spaces for systems. Mathematica provides built-in support for , a prominent subclass of , via commands such as Hypergeometric2F1[a, b, c, z] for the Gauss hypergeometric function and HypergeometricPFQ[{a1, ..., ap}, {b1, ..., bq}, z] for generalized cases. These functions handle symbolic simplification, series expansions around specified points, and high-precision numerical evaluation, leveraging the closure properties of under and . For broader holonomic computations, the HolonomicFunctions package implements algorithms for multivariate cases, including creative telescoping to derive linear relations for integrals and sums. FriCAS and its predecessor OpenAxiom offer domains for algebraic manipulations in D-modules, central to the theory of functions as solutions to systems of linear PDEs. The PartialDifferentialOperator domain constructs rings of partial differential operators over polynomial bases, supporting non-commutative arithmetic and computations for ideals in the Weyl algebra. These features allow users to verify holonomicity by checking module dimensions and perform eliminations on differential ideals, with emphasizing type-safe, category-based implementations for rigorous algebraic verification. Recent developments in include enhancements to the ore_algebra package, with updates around 2019 integrated into versions 9.0 and later, extending support to multivariate operators, enabling computations such as module calculations, operator factorization, and elimination ideals for verifying closure properties. As of 2025, the package continues to be actively maintained, with recent enhancements for numerical evaluation methods in 10.x releases. This open-source implementation facilitates accessible Weyl algebra arithmetic, building on prior algorithms for and operators.