New Identities and Equation Solutions Involving k-Oresme and k-Oresme–Lucas Sequences

Number sequences are among the research areas of interest in both number theory and linear algebra. In particular, the study of matrix representations of recursive sequences is important in revealing the structural properties of these sequences. In this study, the relationships between the elements of the k-Fibonacci and k-Oresme sequences were analyzed using matrix algebra through matrix structures created by connecting the characteristic equations and roots of these sequences. In this context, using the properties of these matrices, the identities A n 2 − A n + 1 A n − 1 = k − 2 n , A n 2 − A n A n − 1 + 1 k 2 A n − 1 2 = k − 2 n , and B n 2 − B n B n − 1 + 1 k 2 B n − 1 2 = − ( k 2 − 4 ) k − 2 n , and some generalizations such as B n + m 2 − ( k 2 − 4 ) A n − t B n + m A t + m − ( k 2 − 4 ) k 2 t − 2 n A t + m 2 = k − 2 m − 2 t B n − t 2 , A t + m 2 − B t − n A n + m A t + m + k 2 n − 2 t A n + m 2 = k − 2 n − 2 m A t − n 2 , and more were derived, where m , n , t ∈ ℤ and t ≠ n . In addition to this, the solution pairs of the algebraic equations x 2 − B p x y + k − 2 p y 2 = k − 2 q A p 2 , x 2 − ( k 2 − 4 ) A p x y − ( k 2 − 4 ) k − 2 p y 2 = k − 2 q B p 2 , and x 2 − B p x y + k − 2 p y 2 = − ( k 2 − 4 ) k − 2 q A p 2 are presented, where A p and B p are k-Oresme and k-Oresme–Lucas numbers, respectively.