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On generalized Fibonacci and Lucas polynomials. (English) Zbl 1198.11017
Summary: Let $h(x)$ be a polynomial with real coefficients. We introduce $h(x)$-Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci polynomials and also the $k$-Fibonacci numbers, and we provide properties for these $h(x)$-Fibonacci polynomials. We also introduce $h(x)$-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix $Q_h(x)$ that generalizes the $Q$-matrix $\left(\matrix 1 & 1 \\ 1 & 0\endmatrix\right)$ whose powers generate the Fibonacci numbers. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:
11B39Fibonacci and Lucas numbers, etc.
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References:
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