On the bi-periodic ( <i>k,s</i> )—Jacobsthal numbers via the Fibonacci–Hörner decomposition of matrix powers

Abstract In this paper we investigate a matrix formulation of the sequence of bi-periodic $(k,s)-$Jacobsthal numbers. The Fibonacci–Hörner and the Jordan form decompositions for computing the matrix powers are used for providing the linear and the combinatorial formulas, as well as the Binet formulas of the bi-periodic $(k,s)-$Jacobsthal numbers. Illustrative special cases are studied.