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arxiv: 2604.04320 · v2 · submitted 2026-04-05 · 🧮 math.FA

A Combinatorial Formula for Recursive Operator Sequences and Applications

Raul E. Curto , Abderrazzak Ech-charyfy , Kaissar Idrissi , El Hassan Zerouali This is my paper

Pith reviewed 2026-05-14 21:21 UTC · model grok-4.3

classification 🧮 math.FA
keywords operator sequenceslinear recurrencecombinatorial formulacommuting operatorsBinet formulaBell polynomialscompanion matrixmoment problem
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The pith

Operator sequences satisfying linear recurrences with commuting coefficients have an explicit combinatorial formula.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes an explicit combinatorial formula for the terms of a sequence of bounded operators on a Hilbert space that obey a linear recurrence with pairwise commuting coefficient operators. Such sequences arise naturally in the operator-valued moment problem and the analysis of flat extensions of block Hankel operators. The formula allows direct calculation of each T_n rather than building it recursively. In the scalar case it specializes to a Binet-type formula involving Bell polynomials that computes powers and exponentials of algebraic operators explicitly. The work also gives a closed expression for the powers of an operator-valued companion matrix.

Core claim

We derive an explicit combinatorial formula for T_n where the sequence satisfies T_{n+r} = A_0 T_n + A_1 T_{n+1} + ⋯ + A_{r-1} T_{n+r-1} for commuting bounded operators A_k. This formula is then applied to express powers of the operator-valued companion matrix, and in the scalar case yields a Binet-type formula using Bell polynomials for algebraic operators.

What carries the argument

The explicit combinatorial formula for T_n in terms of products of the commuting coefficients A_k.

Load-bearing premise

The coefficient operators A_0, A_1, ..., A_{r-1} must be pairwise commuting.

What would settle it

Compute the first few T_n using the recurrence for a specific set of commuting operators and check whether they match the values given by the combinatorial formula.

read the original abstract

We study sequences of bounded operators \((T_n)_{n \ge 0}\) on a complex separable Hilbert space \(\mathcal{H}\) that satisfy a linear recurrence relation of the form $$ T_{n+r} = A_0 T_n + A_1 T_{n+1} + \cdots + A_{r-1} T_{n+r-1} \quad(\textrm{for all } n\ge 0), $$ where the coefficients \(A_0, A_1, \dots, A_{r-1}\) are pairwise commuting bounded operators on \(\mathcal{H}\). \ Such relations naturally arise in the context of the operator-valued moment problem, particularly in the study of flat extensions of block Hankel operators. \ Our first goal is to derive an explicit combinatorial formula for \(T_n\). As a concrete application, we provide an explicit expression for the powers of an operator-valued companion matrix. \ In the special case of scalar coefficients $A_k=a_kI_\mathcal{H}$, with $a_k\in\mathbb{R}$, we recover a Binet-type formula that allows the explicit computation of the powers and the exponential of algebraic operators in terms of Bell polynomials.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper derives an explicit combinatorial formula for the terms T_n of a sequence of bounded operators on a Hilbert space satisfying the linear recurrence T_{n+r} = A_0 T_n + ... + A_{r-1} T_{n+r-1} (n ≥ 0), under the assumption that the coefficients A_k are pairwise commuting bounded operators. It applies the formula to obtain an explicit expression for powers of an operator-valued companion matrix and recovers a Binet-type formula involving Bell polynomials when the coefficients are scalar multiples of the identity.

Significance. If the derivation holds, the result supplies a closed-form combinatorial expression for recursive operator sequences that avoids iterative application of the recurrence. This is potentially useful in the study of operator-valued moment problems and flat extensions of block Hankel operators. The recovery of the scalar Binet formula with Bell polynomials provides a useful consistency check, and the left-acting structure under commutativity avoids ordering ambiguities.

minor comments (3)
  1. [§2] §2: The combinatorial polynomials P_{n,k} are introduced via generating functions but their explicit multi-index form is not written out; adding the sum-over-partitions expression would make the formula immediately usable without further derivation.
  2. [§4] The application to the companion matrix in §4 assumes the initial block is the identity; a brief remark on the general initial data case would clarify the scope.
  3. [Abstract] The abstract claims the formula is 'parameter-free' in the commuting case, but the dependence on the initial T_0,...,T_{r-1} is implicit; rephrasing avoids potential misreading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript on the combinatorial formula for recursive operator sequences. The recommendation for minor revision is noted, but the report lists no specific major comments or requested changes. We are pleased that the utility for operator-valued moment problems and the recovery of the scalar Binet formula are viewed favorably. If any unlisted minor editorial points exist, we will address them promptly in revision.

Circularity Check

0 steps flagged

No significant circularity; derivation follows directly from recurrence via induction and commutativity

full rationale

The central claim is an explicit combinatorial formula for T_n obtained by unfolding the given linear recurrence T_{n+r} = sum A_k T_{n+k} under the pairwise commutativity assumption on the A_k. Commutativity permits treating the coefficients as elements of a commutative algebra, so the expansion T_n = sum P_{n,k}(A_0,...,A_{r-1}) T_k with combinatorial polynomials P follows by direct induction on n or by generating-function algebra; no parameter is fitted to data and then re-labeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The scalar case recovers the known Bell-polynomial expression for Binet-type formulas as a special case, confirming the derivation is self-contained and independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the recurrence coefficients commute, which is a standard domain assumption in operator theory but is essential for the formula to hold without additional correction terms.

axioms (1)
  • domain assumption The coefficients A_0 through A_{r-1} are pairwise commuting bounded operators on H.
    Invoked in the statement of the recurrence and required for the combinatorial formula to be well-defined without ordering issues.

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Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Arabian Journal of Mathematics,10(3) (2021), 555–574

    Benkhaldoun, H., Ben Taher, R., Rachidi, M.:Periodic matrix difference equations and companion matrices in blocks: some applications. Arabian Journal of Mathematics,10(3) (2021), 555–574

    work page 2021

  2. [2]

    B., Rachidi, M., Zerouali, E

    Bentaher, R. B., Rachidi, M., Zerouali, E. H.:Recursive Subnormal Completion and the Truncated Moment Problem. Bull. London Math. Soc.,33(4) (2001), 425–432

    work page 2001

  3. [3]

    B., Rachidi, M.:Linear Recurrence Relations in the Algebra of Matrices and Applications

    Bentaher, R. B., Rachidi, M.:Linear Recurrence Relations in the Algebra of Matrices and Applications. Linear Algebra Appl.,330(1–3) (2001), 15–24

    work page 2001

  4. [4]

    Taher, R.B. and Rachidi, M.:On the matrix powers and exponential by the r-generalized Fi- bonacci sequences methods: the companion matrix case, Linear Algebra and Its Applications, 370(2003), 341–353

    work page 2003

  5. [5]

    Applied Mathematics and Computation,290(2016), 267–280

    Bentaher, R.B, and Rachidi, M.:Solving some generalized Vandermonde systems and inverse of their associate matrices via new approaches for the Binet formula. Applied Mathematics and Computation,290(2016), 267–280

    work page 2016

  6. [6]

    Chen, W. Y. C., Louck, J. D.:The combinatorial power of the companion matrix. Linear Algebra and its Applications,232(1996), 261–278

    work page 1996

  7. [7]

    E., Rachidi, M

    Chidume, C. E., Rachidi, M. and Zerouali, E. H.:Solving the general truncated moment problem by ther-generalized Fibonacci sequences method. Journal of Mathematical Analysis and Applications,256(2), 625–635 (2001). COMBINATORIAL FORMULA FOR OPERATOR SEQUENCES 17

    work page 2001

  8. [8]

    Springer Science & Business Media, New York (2012)

    Comtet, L.:Advanced Combinatorics: The Art of Finite and Infinite Expansions. Springer Science & Business Media, New York (2012)

    work page 2012

  9. [9]

    E., Ech-charyfy, A., El Azhar, H., Zerouali, E

    Curto, R. E., Ech-charyfy, A., El Azhar, H., Zerouali, E. H.:The Local Operator Moment Problem onR. Complex Analysis and Operator Theory,19(2) (2025), 25

    work page 2025

  10. [10]

    Preprint (2025)

    Curto, R.E., Ech-charyfy, A., Idrissi, K., H., Zerouali, E.H.:Infinite-dimensional flat exten- sions in operator moment problems. Preprint (2025)

    work page 2025

  11. [11]

    Preprint (2023)

    Curto, R.E., Ech-charyfy, A., Idrissi, K., Zerouali, E.H.:A Recursive approach to the matrix moment problem. Preprint (2023)

    work page 2023

  12. [12]

    and Fialkow, L.A.:Recursiveness, positivity and truncated moment problems

    Curto, R.E. and Fialkow, L.A.:Recursiveness, positivity and truncated moment problems. Houston Journal of Mathematics, 17, 603–635 (1991)

    work page 1991

  13. [13]

    E., Fialkow, L

    Curto, R. E., Fialkow, L. A.:Solution of the truncated complex moment problem for flat data.Mem. Amer. Math. Soc. 119(568), x+52 pp. (1996)

    work page 1996

  14. [14]

    Curto, R.E., Fialkow, L.A.:Flat extensions of positive moment matrices: Recursively gen- erated relations.Mem. Amer. Math. Soc. 136(648), x+56 pp. (1998)

    work page 1998

  15. [15]

    R.:What Does the Spectral Theorem Say?The American Mathematical Monthly, 70(3) (1963), 241–247

    Halmos, P. R.:What Does the Spectral Theorem Say?The American Mathematical Monthly, 70(3) (1963), 241–247

    work page 1963

  16. [16]

    Mourrain, B., Schm¨ udgen, K.:Flat extensions in∗-algebras.Proc. Amer. Math. Soc. 144(11), 4873–4885 (2016)

    work page 2016

  17. [17]

    P.:An operator-valued generalization of Tchakaloff’s Theorem.J

    Kimsey, D. P.:An operator-valued generalization of Tchakaloff’s Theorem.J. Funct. Anal. 266(3), 1170–1184 (2014)

    work page 2014

  18. [18]

    P., Trachana, M.:On a solution of the multidimensional truncated matrix-valued moment problem.Milan J

    Kimsey, D. P., Trachana, M.:On a solution of the multidimensional truncated matrix-valued moment problem.Milan J. Math. 90(1), 17–101 (2022)

    work page 2022

  19. [19]

    The Fibonacci Quarterly,23(4) (1985), 290–295

    Levesque, C.:Onmth Order Linear Recurrences. The Fibonacci Quarterly,23(4) (1985), 290–295

    work page 1985

  20. [20]

    Mez˝ o, I.:Ther-Bell numbers. J. Integer Seq.14(1), 1–14 (2011)

    work page 2011

  21. [21]

    The Fibonacci Quarterly,37(1999), 34–38

    Mouline, M., Rachidi, M.:Application of Markov Chains Properties tor-Generalized Fi- bonacci Sequences. The Fibonacci Quarterly,37(1999), 34–38

    work page 1999

  22. [22]

    In: Bergum, G.E., Philippou, A.N., Horadam, A.F

    Philippou, G.N.:On thekth Order Linear Recurrence and Some Probability Applications. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds.),Applications of Fibonacci Numbers. Kluwer Academic Publishers, Dordrecht (1988)

    work page 1988

  23. [23]

    Birkh¨ auser, Progress in Mathematics, Vol

    Schm¨ udgen, K.:Unbounded Operator Algebras and Representation Theory. Birkh¨ auser, Progress in Mathematics, Vol. 37 (2013)

    work page 2013

  24. [24]

    Tchakaloff, V.:Formules de cubatures m´ ecaniques ` a coefficients non n´ egatifs.Bull. Sci. Math 81, no. 2, 123–134 (1957). Department of Mathematics, The University of Iowa, Iowa City, Iowa, U.S.A. Email address:raul-curto@uiowa.edu Laboratory of Mathematical Analysis and Applications,, F aculty of Sciences, Mo- hammed V University in Rabat,, Rabat, Mor...

    work page 1957