In this paper, we considered the generalized bi-periodic Fibonacci polynomials, and obtained some identities related to generalized bi-periodic Fibonacci polynomials using the matrix theory. In addition, the generalized bi-periodic Lucas polynomial was defined by $ L_{n}\left (x \right) = bp\left (x \right) L_{n-1}\left (x \right)+q\left (x \right)L_{n-2}\left (x \right) $ (if $ n $ is even) or $ L_{n}\left (x \right) = ap\left (x \right) L_{n-1}\left (x \right)+q\left (x \right)L_{n-2}\left (x \right) $ (if $ n $ is odd), with initial conditions $ L_{0}\left (x \right) = 2 $, $ L_{1}\left (x \right) = ap\left (x \right) $, where $ p\left (x \right) $ and $ q\left (x \right) $ were nonzero polynomials in $ Q \left [ x \right ] $. We obtained a series of identities related to the generalized bi-periodic Fibonacci and Lucas polynomials.
Citation: Tingting Du, Zhengang Wu. Some identities of the generalized bi-periodic Fibonacci and Lucas polynomials[J]. AIMS Mathematics, 2024, 9(3): 7492-7510. doi: 10.3934/math.2024363
In this paper, we considered the generalized bi-periodic Fibonacci polynomials, and obtained some identities related to generalized bi-periodic Fibonacci polynomials using the matrix theory. In addition, the generalized bi-periodic Lucas polynomial was defined by $ L_{n}\left (x \right) = bp\left (x \right) L_{n-1}\left (x \right)+q\left (x \right)L_{n-2}\left (x \right) $ (if $ n $ is even) or $ L_{n}\left (x \right) = ap\left (x \right) L_{n-1}\left (x \right)+q\left (x \right)L_{n-2}\left (x \right) $ (if $ n $ is odd), with initial conditions $ L_{0}\left (x \right) = 2 $, $ L_{1}\left (x \right) = ap\left (x \right) $, where $ p\left (x \right) $ and $ q\left (x \right) $ were nonzero polynomials in $ Q \left [ x \right ] $. We obtained a series of identities related to the generalized bi-periodic Fibonacci and Lucas polynomials.
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Tingting Du, Zhengang Wu. Some identities of the generalized bi-periodic Fibonacci and Lucas polynomials[J]. AIMS Mathematics, 2024, 9(3): 7492-7510. doi: 10.3934/math.2024363
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