Variants of the Filbert Matrix
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A variation of the Filbert matrix from  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.