Closed forms for powers and inverses of special matrices

TL;DR

This paper derives explicit formulas for powers and inverses of certain special matrices, using advanced combinatorial tools like Lagrange inversion and Bell polynomials, with applications to combinatorial sequences.

Contribution

It provides new closed-form expressions for powers and inverses of specific upper triangular and non-triangular matrices, expanding the understanding of their algebraic properties.

Findings

Explicit formulas for matrix powers and inverses are derived.

The methods involve Lagrange inversion and Bell polynomials.

Applications to combinatorial sequences are discussed.

Abstract

This contribution is motivated by old and recent works on matrix powers and their applications on combinatorial sequences. We give in this paper the $s$-th powers and the inverses for special upper triangular matrices and the $s$-th powers for special non-triangular matrices. The used tools are Lagrange inversion formula and the partial Bell polynomials.