Abstract
In the present article, we investigate the properties of bivariate Fibonacci polynomials of order k in terms of the generating functions. For k and ℓ (1 ≤ ℓ ≤ k − 1), the relationship between the bivariate Fibonacci polynomials of order k and the bivariate Fibonacci polynomials of order ℓ is elucidated. Lucas polynomials of order k are considered. We also reveal the relationship between Lucas polynomials of order k and Lucas polynomials of order ℓ. The present work extends several properties of Fibonacci and Lucas polynomials of order k, which will lead us a new type of geneses of these polynomials. We point out that Fibonacci and Lucas polynomials of order k are closely related to distributions of order k and show that the distributions possess properties analogous to the bivariate Fibonacci and Lucas polynomials of order k.
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This research was partially supported by the ISM Cooperative Research Program (2008-ISM·CRP-2009).
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Inoue, K., Aki, S. Bivariate Fibonacci polynomials of order k with statistical applications. Ann Inst Stat Math 63, 197–210 (2011). https://doi.org/10.1007/s10463-008-0217-x
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DOI: https://doi.org/10.1007/s10463-008-0217-x