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Variants of the Filbert matrix. (English) Zbl 1306.11019

Summary: A variation of the Filbert matrix from the authors’ paper [Fibonacci Q. 48, No. 1, 29–33 (2010; Zbl 1219.11028)] 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 formulas 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.

MSC:

11B39 Fibonacci and Lucas numbers and polynomials and generalizations
05A30 \(q\)-calculus and related topics
15A23 Factorization of matrices

Citations:

Zbl 1219.11028
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