Variants of the Filbert Matrix

Abstract

A variation of the Filbert matrix from [1] is introduced, which has one additional Fibonacci factor in the numerator. We also introduce its Lucas counterpart by taking Lucas numbers instead of Fibonacci numbers in a similar manner. Explicit formulæ are derived for the LU-decompositions, their inverses, the inverse matrix, as well as the Cholesky decompositions. The approach is to use q-analysis and to guess the relevant quantities, and proving them later by induction.