@article{article_1809730, title={A combinatorial formula for recursive operator sequences and applications}, journal={Constructive Mathematical Analysis}, volume={8}, pages={200–216}, year={2025}, DOI={10.33205/cma.1809730}, author={Curto, Raul and Ech-Charyfy, Abderrazzak and Idrissi, Kaissar and Zerouali, El Hassan}, keywords={Operator recurrence relation, operator-valued moment problem, Binet formula, combinatorial formula, companion matrix}, 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.}, number={4}, publisher={Tuncer ACAR}