A nonlinear generalization of the Filbert matrix and its Lucas analogue

In this paper, we present both a new generalization and an analogue of the Filbert matrix (Formula presented.) by the means of the Fibonacci and Lucas numbers whose indices are in nonlinear form (Formula presented.) for the positive integers (Formula presented.) and the integers r, s, c. This will be the first example as nonlinear generalizations of the Filbert and Lilbert matrices. Furthermore, we present q-versions of these matrices and their related results. We derive explicit formulæ for the inverse matrix, the LU-decomposition and the inverse matrices (Formula presented.), (Formula presented.) as well as we present the Cholesky decomposition for all matrices.