Holonomic Ansatz in Symbolic Computation

• Holonomic Ansatz is a method that assumes sequences satisfy linear recurrences with polynomial coefficients and reconstructs these using initial values and linear algebra.
• It uses techniques like Laplace expansion, creative telescoping, and holonomic closure properties to prove complex identities in determinants and Pfaffians.
• Hybrid variants integrate C-finite and Almkvist–Zeilberger algorithms, bridging rational generating functions with differential equations in computer algebra.

The holonomic ansatz is a guess-and-verify methodology in symbolic computation for discovering and proving linear recurrences with polynomial coefficients, and for exploiting such recurrences in determinant, Pfaffian, summation, and generating-function problems. In its sequence form, it starts from the assumption that a sequence is holonomic, or P-recursive, and reconstructs its annihilating recurrence from sufficiently many initial values. In its determinant form, associated with Zeilberger, it augments Laplace expansion by auxiliary cofactor ratios and then proves the required identities by holonomic closure properties and creative telescoping. Subsequent adaptations extend the method to Pfaffians and skew-symmetric matrices, and hybrid variants combine it with the $C$-finite ansatz and continuous Almkvist–Zeilberger algorithms for integral-defined sequences (Krityakierne et al., 2022, Ishikawa et al., 2012).

1. Formal setting and recurrence-theoretic basis

A univariate sequence $(f_n)$ is holonomic if it satisfies a linear recurrence with polynomial coefficients,

$$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$

or equivalently, if an operator

$T(N):=\sum_{i=0}^r p_i(n)\,N^i$

annihilates the sequence, where $N$ is the forward-shift operator $N\,a_n=a_{n+1}$. In the formulation used for sequence guessing, the pair $(r,d)$, with $r$ the order and $d=\max_i\deg p_i$, is called the type of the holonomic recurrence. The same literature also uses the terminology holonomic, P-recursive, and, in the generating-function setting, D-finite (Krityakierne et al., 2022, Thanatipanonda et al., 2020).

The ansatz assumes a priori bounds on the order and degree. One writes

$p_i(n)=c_{i,0}+c_{i,1}n+\cdots+c_{i,k}n^k$

for fixed $(f_n)$0, substitutes into

$(f_n)$1

and enforces the relation for

$(f_n)$2

This yields a homogeneous linear system $(f_n)$3 of size $(f_n)$4. If sufficiently many initial terms are known and the sequence is indeed holonomic of order at most $(f_n)$5 and degree at most $(f_n)$6, reduced-row-echelon-form or null-space computation yields the polynomial coefficients $(f_n)$7. Verification for all further terms is then required; if the recurrence fails, one increases $(f_n)$8 (Krityakierne et al., 2022).

A central structural feature is the relation between recurrences and generating functions. If

$(f_n)$9

and $$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$0 is holonomic, then $$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$1 satisfies a linear differential equation with polynomial coefficients; conversely, a formal power series satisfying such a differential equation has holonomic coefficients. This recurrence–ODE duality is one reason the ansatz integrates naturally with symbolic summation and differential-algebraic algorithms (Krityakierne et al., 2022, Thanatipanonda et al., 2020).

2. Zeilberger’s determinant framework

Zeilberger’s original holonomic ansatz for determinant evaluations starts with a family

$$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$2

and applies Laplace expansion along the last row: $$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$3 where

$$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$4

The key auxiliary quantities are the normalized cofactors

$$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$5

They satisfy three identities for $$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$6: $$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$7

$$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$8

and

$$p_0(n)f_{n} + p_1(n)f_{n+1} + \dots + p_r(n)f_{n+r} = 0,$$9

The determinant problem is thereby reduced to proving that the bivariate auxiliary sequence is holonomic and that these identities hold (Ishikawa et al., 2012).

The operational outline is a characteristic “guess–prove–substitute” cycle. One computes enough values of $T(N):=\sum_{i=0}^r p_i(n)\,N^i$0, uses a guessing package to conjecture recurrences and initial values, proves the conjectured recurrences and the three defining identities by creative telescoping and holonomic closure properties, and finally substitutes the conjectured ratio $T(N):=\sum_{i=0}^r p_i(n)\,N^i$1 into the last identity to close the induction. In determinant problems with holonomic entries, this turns evaluation into a structured elimination problem in the holonomic systems framework (Ishikawa et al., 2012).

The approach has also been adapted to determinants with shifted binomial coefficients and signed Kronecker deltas. In that setting, elementary Pascal-shift factors

$T(N):=\sum_{i=0}^r p_i(n)\,N^i$2

are used to simplify the matrix via

$T(N):=\sum_{i=0}^r p_i(n)\,N^i$3

After these shifts, the top-left corner becomes sparse and the bottom-right block remains in the same family with shifted parameters. The ansatz is then applied along the first column rather than the last row by defining suitable cofactor ratios of that column. For negative deltas, a formal perturbation

$T(N):=\sum_{i=0}^r p_i(n)\,N^i$4

avoids an indeterminate $T(N):=\sum_{i=0}^r p_i(n)\,N^i$5 situation and produces modified linear systems for the new cofactor ratios $T(N):=\sum_{i=0}^r p_i(n)\,N^i$6 (Du et al., 2021).

3. Pfaffian adaptation and skew-symmetric matrices

A major extension addresses Pfaffians, motivated by the fact that if $T(N):=\sum_{i=0}^r p_i(n)\,N^i$7 is $T(N):=\sum_{i=0}^r p_i(n)\,N^i$8 skew-symmetric, then

$T(N):=\sum_{i=0}^r p_i(n)\,N^i$9

The determinant ansatz breaks down for skew-symmetric matrices of odd size because the determinant is $N$0, and the original approach also requires nonsingularity and holonomicity of the cofactor ratio. The Pfaffian variant replaces Laplace expansion by the cofactor expansion of the Pfaffian and thereby applies directly to skew-symmetric settings for which the determinant formulation does not work (Ishikawa et al., 2012).

For a $N$1 skew-symmetric matrix $N$2, the Pfaffian is defined by

$N$3

where the sum runs over perfect matchings of $N$4. The relevant cofactors are

$N$5

and they satisfy

$N$6

The Pfaffian holonomic ansatz defines

$N$7

and proves the three identities

$N$8

$N$9

and

$N\,a_n=a_{n+1}$0

From the last relation and induction one concludes $N\,a_n=a_{n+1}$1 explicitly (Ishikawa et al., 2012).

This adaptation is significant because it turns Pfaffian evaluation into the same kind of routine holonomic workflow available in the determinant case. The paper explicitly states that many Pfaffian evaluations arising, for instance, from plane-partition enumerations, dimer-model determinants via Kasteleyn’s method, and Hankel-type Pfaffians can now be attacked in a fully routine way, and that determinants of skew-symmetric matrices are covered as a consequence of the identity $N\,a_n=a_{n+1}$2 (Ishikawa et al., 2012).

4. Algorithmic realization and computer algebra

The computational core of the holonomic ansatz is linear algebra plus closure theory. In the sequence setting, the unknown coefficients $N\,a_n=a_{n+1}$3 are recovered from a finite linear system. In determinant and Pfaffian settings, one first generates numerical data for auxiliary cofactor ratios by direct determinant or Pfaffian expansion, or by solving the defining linear systems for the cofactors, and then applies guessing software to infer recurrences in the discrete parameters (Krityakierne et al., 2022, Du et al., 2021).

The proof stage relies on holonomic systems machinery. The cited implementations include Kauers’ Guess package, Koutschan’s HolonomicFunctions package, Maple’s gfun, and routines such as GuessHo and HoToDiff. In the determinant and Pfaffian literature, creative telescoping is used to eliminate summation indices from expressions such as

$N\,a_n=a_{n+1}$4

or the boundary sum

$N\,a_n=a_{n+1}$5

producing recurrences for the telescoped sums. Closure under finite sums, products, diagonals, convolution, and partial sums is essential to this step, as is the Ore-algebra representation of recurrences via shift operators satisfying commutation rules such as $N\,a_n=a_{n+1}$6 (Du et al., 2021, Krityakierne et al., 2022).

A hybrid algorithmic line combines the $N\,a_n=a_{n+1}$7-finite ansatz with holonomic techniques. For sequences defined by integrals of powers of $N\,a_n=a_{n+1}$8-finite polynomial sequences,

$N\,a_n=a_{n+1}$9

one forms the generating function

$(r,d)$0

where $(r,d)$1 is rational by the $(r,d)$2-finite ansatz. The continuous Almkvist–Zeilberger algorithm then finds a differential operator $(r,d)$3 and a certificate $(r,d)$4 such that

$(r,d)$5

Integration yields an inhomogeneous differential equation for $(r,d)$6, which is converted into a holonomic recurrence for $(r,d)$7. This establishes a direct bridge between rational generating functions, definite integrals, D-finite differential equations, and P-recursive sequences (Ekhad et al., 2015).

5. Representative evaluations and applications

The Pfaffian version was demonstrated on conjectures from “Pfaffian decomposition and a Pfaffian analogue of $(r,d)$8-Catalan Hankel determinants.” For the Motzkin numbers $(r,d)$9, the paper proves

$r$0

For the central Delannoy numbers

$r$1

it proves

$r$2

For the Narayana polynomials

$r$3

it proves

$r$4

with the sign issue handled by polynomial-identity arguments. A minor summation formula related to partitions and Motzkin paths follows as a corollary (Ishikawa et al., 2012).

In tiling theory, the ansatz was used for binomial determinants with signed Kronecker deltas located along arbitrary diagonals. These determinants count cyclically symmetric rhombus tilings of hexagonal regions with triangular holes via the Lindström–Gessel–Viennot theorem. By adapting the ansatz and combining it with computer algebra, the authors obtained closed forms for all previously unresolved conjectures in the family, including a 2005 conjecture of Lascoux and Krattenthaler and conjectures 20, 21, and 24 of Koutschan–Thanatipanonda. The paper also establishes determinant identities such as switching lemmas in $r$5, “triangle” relations between neighboring determinant families, and combinatorial non-vanishing statements (Du et al., 2021).

At the level of elementary sequence recognition, the ansatz recovers standard recurrences such as

$r$6

for Catalan numbers,

$r$7

for harmonic numbers in the notation of the cited guide, and

$r$8

for $r$9. These examples illustrate the general procedure of fixing $d=\max_i\deg p_i$0, solving the coefficient system, and then translating the recurrence into a differential equation for the generating function when needed (Krityakierne et al., 2022).

6. Scope, limitations, and related uses of “holonomic”

The principal limitation of the holonomic ansatz is not conceptual but algorithmic. The size of the linear system in the sequence setting is $d=\max_i\deg p_i$1, so practical use requires that $d=\max_i\deg p_i$2 and $d=\max_i\deg p_i$3 remain small. The literature explicitly emphasizes that knowing a priori bounds on order and degree, from theory or closure-property considerations, is essential to make the guess rigorous. Large reduced-row-echelon-form systems, Gröbner-basis manipulations, and telescoping computations can become expensive, and some of the determinant calculations reported in the tiling work took on the order of hours to days on a modern server (Krityakierne et al., 2022, Du et al., 2021).

The ansatz also sits inside a hierarchy of related recurrence paradigms. Polynomial sequences and $d=\max_i\deg p_i$4-finite sequences are special cases of holonomic sequences. The 2022 survey places the holonomic ansatz alongside polynomial, $d=\max_i\deg p_i$5-finite, and $d=\max_i\deg p_i$6-finite ansatzes, while the 2020 note introduces the broader $d=\max_i\deg p_i$7-recursive ansatz, in which the recurrence coefficients are themselves $d=\max_i\deg p_i$8-finite sequences. The trade-off is explicit: holonomic sequences retain strong closure properties and efficient guessing algorithms, whereas $d=\max_i\deg p_i$9-recursive sequences form a strictly larger class but lead to non-linear systems and harder elimination problems (Krityakierne et al., 2022, Thanatipanonda et al., 2020).

A recurrent terminological confusion is the relation between the holonomic ansatz of symbolic computation and holonomic quantum computation. The latter concerns geometric phases, Wilczek–Zee connections, and holonomies of degenerate quantum subspaces, as in Weyl disks, NV-center $p_i(n)=c_{i,0}+c_{i,1}n+\cdots+c_{i,k}n^k$0-systems, and photonic $p_i(n)=c_{i,0}+c_{i,1}n+\cdots+c_{i,k}n^k$1-pods. In those works, “holonomic” refers to geometric manipulation of quantum states through Abelian or non-Abelian Berry connections, not to P-recursive recurrences, closure properties, or creative telescoping. The shared adjective reflects the notion of holonomy, but the mathematical content and algorithmic objectives are different (Boogers et al., 2021, Zhou et al., 2017, Pinske et al., 2020).

The modern mathematical meaning of the holonomic ansatz is therefore best understood as a unifying framework for exact guessing and certification. It links recurrences, differential equations, determinant and Pfaffian expansions, generating functions, and symbolic summation into a common machine-assisted proof strategy. Within that framework, the adaptation to Pfaffians, the treatment of signed-delta determinants from tiling theory, and the interaction with $p_i(n)=c_{i,0}+c_{i,1}n+\cdots+c_{i,k}n^k$2-finite and integral methods show that the ansatz is not a single algorithm but a family of tightly related procedures organized by the holonomic systems viewpoint (Ishikawa et al., 2012, Ekhad et al., 2015).