A combinatorial formula for recursive operator sequences and applications

Year 2025, Volume: 8 Issue: 4, 200 - 216, 15.12.2025

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

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

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

Operator recurrence relation , operator-valued moment problem , Binet formula , combinatorial formula , companion matrix

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.