Note on the Chebyshev Polynomials and Applications to the Fibonacci Numbers

José Morgado

Abstract: In [12], Gheorghe Udrea generalizes a result obtained in [8], by showing that, if $(U_{n})_{n\ge0}$ is the sequence of Chebyshev polynomials of the second kind, then the product of any two distinct elements of the set $$ \Bigl\{U_{n},\,U_{n+2r},\,U_{n+4r},\,4U_{n+r}U_{n+2r}U_{n+3r}\Bigr\}, r,n\in\N, $$ increased by $U_{a}^{2}U_{b}^{2}$, for suitable nonnegative integers $a$ and $b$, is a perfect square.
In this note, one obtains a similar result for the Chebyshev polynomials of the first kind and one states some generalizations of results contained in [12] and in [8].

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