A combinatorial formula for recursive operator sequences and applications

Raul Curto; Abderrazzak Ech-Charyfy; Kaissar Idrissi; El Hassan Zerouali

doi:10.33205/cma.1809730

A combinatorial formula for recursive operator sequences and applications

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.

Keywords

References

H. Benkhaldoun, R. Ben Taher and M. Rachidi: Periodic matrix difference equations and companion matrices in blocks: some applications, Arab. J. Math., 10 (3) (2021), 555–574.
R. B. Bentaher, M. Rachidi and E. H. Zerouali: Recursive Subnormal Completion and the Truncated Moment Problem, Bull. London Math. Soc., 33 (4) (2001), 425–432.
R. B. Bentaher, M. Rachidi: Linear Recurrence Relations in the Algebra of Matrices and Applications, Linear Algebra Appl., 330 (1–3) (2001), 15–24.
R. B. Taher, M. Rachidi: On the matrix powers and exponential by the r-generalized Fibonacci sequences methods: the companion matrix case, Linear Algebra Appl., 370 (2003), 341–353.
R. B. Bentaher, M. Rachidi: Solving some generalized Vandermonde systems and inverse of their associate matrices via new approaches for the Binet formula, Appl. Math. Comput., 290 (2016), 267–280.
W. Y. C. Chen, J. D. Louck: The combinatorial power of the companion matrix, Linear Algebra Appl., 232 (1996), 261–278.
C. E. Chidume, M. Rachidi and E. H. Zerouali: Solving the general truncated moment problem by the r-generalized Fibonacci sequences method, J. Math. Anal. Appl., 256 (2) (2001), 625–635.
L. Comtet: Advanced Combinatorics: The Art of Finite and Infinite Expansions, Springer Science & Business Media, New York (2012).

R. E. Curto, A. Ech-charyfy, H. El Azhar and E. H. Zerouali: The Local Operator Moment Problem on R, Complex Anal. Oper. Theory, 19 (2) (2025), Article ID: 25.
R. E. Curto, A. Ech-charyfy, K. H. Idrissi and E. H. Zerouali: Infinite-dimensional flat extensions in operator moment problems, Preprint (2025).
R. E. Curto, A. Ech-charyfy, K. Idrissi and E. H. Zerouali: A Recursive approach to the matrix moment problem, Preprint (2023).
R. E. Curto, L. A. Fialkow: Recursiveness, positivity and truncated moment problems, Houston J. Math., 17 (1991), 603–635.
R. E. Curto, L. A. Fialkow: Solution of the truncated complex moment problem for flat data, Mem. Amer. Math. Soc., 119 (568) (1996).
R. E. Curto, L. A. Fialkow: Flat extensions of positive moment matrices: Recursively generated relations, Mem. Amer. Math. Soc., 136 (648) (1998).
P. R. Halmos: What Does the Spectral Theorem Say?, The Amer. Math. Monthly, 70 (3) (1963), 241–247.
B. Mourrain, K. Schmüdgen: Flat extensions in ∗-algebras, Proc. Amer. Math. Soc., 144 (11), 4873–4885 (2016).
D. P. Kimsey: An operator-valued generalization of Tchakaloff’s Theorem, J. Funct. Anal., 266 (3) (2014), 1170–1184.
D. P. Kimsey, M. Trachana: On a solution of the multidimensional truncated matrix-valued moment problem, Milan J. Math., 90 (1) (2022), 17–101.
C. Levesque: On mth Order Linear Recurrences, Fibonacci Quart., 23 (4) (1985), 290–295.
I. Mez˝o: The r-Bell numbers, J. Integer Seq., 14 (1) (2011), Article ID: 11.1.1.
M. Mouline, M. Rachidi: Application of Markov Chains Properties to r-Generalized Fibonacci Sequences, Fibonacci Quart., 37 (1999), 34–38.
G. N. Philippou: On the kth 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).
K. Schmüdgen: Unbounded Operator Algebras and Representation Theory, vol. 37, Birkhäuser, Progress in Mathematics (2013).
V. Tchakaloff: Formules de cubatures mécaniques à coefficients non négatifs, Bull. Sci. Math., 81 (2) (1957), 123–134.

Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis

Journal Section

Research Article

Authors

Raul Curto ^*
0000-0002-1776-5080
United States

Abderrazzak Ech-Charyfy
0009-0004-8273-5496
Morocco

Kaissar Idrissi
0000-0002-3279-4613
Morocco

El Hassan Zerouali
0000-0001-6240-7859
Morocco

Early Pub Date

December 10, 2025

Publication Date

December 15, 2025

Submission Date

October 24, 2025

Acceptance Date

December 2, 2025

Published in Issue

Year 2025 Volume: 8 Number: 4

DOI

https://doi.org/10.33205/cma.1809730

IZ

https://izlik.org/JA98AR65UF

Cite

RIS / Bibtex

APA

Curto, R., Ech-Charyfy, A., Idrissi, K., & Zerouali, E. H. (2025). A combinatorial formula for recursive operator sequences and applications. Constructive Mathematical Analysis, 8(4), 200-216. https://doi.org/10.33205/cma.1809730

AMA

1.Curto R, Ech-Charyfy A, Idrissi K, Zerouali EH. A combinatorial formula for recursive operator sequences and applications. CMA. 2025;8(4):200-216. doi:10.33205/cma.1809730

Chicago

Curto, Raul, Abderrazzak Ech-Charyfy, Kaissar Idrissi, and El Hassan Zerouali. 2025. “A Combinatorial Formula for Recursive Operator Sequences and Applications”. Constructive Mathematical Analysis 8 (4): 200-216. https://doi.org/10.33205/cma.1809730.

EndNote

Curto R, Ech-Charyfy A, Idrissi K, Zerouali EH (December 1, 2025) A combinatorial formula for recursive operator sequences and applications. Constructive Mathematical Analysis 8 4 200–216.

IEEE

[1]R. Curto, A. Ech-Charyfy, K. Idrissi, and E. H. Zerouali, “A combinatorial formula for recursive operator sequences and applications”, CMA, vol. 8, no. 4, pp. 200–216, Dec. 2025, doi: 10.33205/cma.1809730.

ISNAD

Curto, Raul - Ech-Charyfy, Abderrazzak - Idrissi, Kaissar - Zerouali, El Hassan. “A Combinatorial Formula for Recursive Operator Sequences and Applications”. Constructive Mathematical Analysis 8/4 (December 1, 2025): 200-216. https://doi.org/10.33205/cma.1809730.

JAMA

1.Curto R, Ech-Charyfy A, Idrissi K, Zerouali EH. A combinatorial formula for recursive operator sequences and applications. CMA. 2025;8:200–216.

MLA

Curto, Raul, et al. “A Combinatorial Formula for Recursive Operator Sequences and Applications”. Constructive Mathematical Analysis, vol. 8, no. 4, Dec. 2025, pp. 200-16, doi:10.33205/cma.1809730.

Vancouver

1.Raul Curto, Abderrazzak Ech-Charyfy, Kaissar Idrissi, El Hassan Zerouali. A combinatorial formula for recursive operator sequences and applications. CMA. 2025 Dec. 1;8(4):200-16. doi:10.33205/cma.1809730