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Generalized Humbert polynomials via generalized Fibonacci polynomials

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  • Wang, Weiping
  • Wang, Hui

Abstract

In this paper, we define the generalized (p, q)-Fibonacci polynomials un, m(x) and generalized (p, q)-Lucas polynomials vn, m(x), and further introduce the generalized Humbert polynomials un,m(r)(x) as the convolutions of un, m(x). We give many expressions, expansions, recurrence relations and differential recurrence relations of un,m(r)(x), and study the matrices and determinants related to the polynomials un, m(x), vn, m(x) and un,m(r)(x). Finally, we present an algebraic interpretation for the generalized Humbert polynomials un,m(r)(x). It can be found that various well-known polynomials are special cases of un, m(x), vn, m(x) or un,m(r)(x). Therefore, by introducing the general polynomials un, m(x), vn, m(x) and un,m(r)(x), we have a unified approach to dealing with many special polynomials in the literature.

Suggested Citation

  • Wang, Weiping & Wang, Hui, 2017. "Generalized Humbert polynomials via generalized Fibonacci polynomials," Applied Mathematics and Computation, Elsevier, vol. 307(C), pages 204-216.
  • Handle: RePEc:eee:apmaco:v:307:y:2017:i:c:p:204-216
    DOI: 10.1016/j.amc.2017.02.050
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    References listed on IDEAS

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    1. Nalli, Ayse & Haukkanen, Pentti, 2009. "On generalized Fibonacci and Lucas polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3179-3186.
    2. Djordjević, Gospava B. & Djordjević, Snežana S., 2015. "Convolutions of the generalized Morgan–Voyce polynomials," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 106-115.
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    Cited by:

    1. Yuankui Ma & Wenpeng Zhang, 2018. "Some Identities Involving Fibonacci Polynomials and Fibonacci Numbers," Mathematics, MDPI, vol. 6(12), pages 1-8, December.
    2. Florek, Wojciech, 2018. "A class of generalized Tribonacci sequences applied to counting problems," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 809-821.

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