Displaying similar documents to “Note on the Chebyshev polynomials and applications to the Fibonacci numbers.”

Circulants and the factorization of the Fibonacci–like numbers

Jaroslav Seibert, Pavel Trojovský (2006)

Acta Mathematica Universitatis Ostraviensis

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Several authors gave various factorizations of the Fibonacci and Lucas numbers. The relations are derived with the help of connections between determinants of tridiagonal matrices and the Fibonacci and Lucas numbers using the Chebyshev polynomials. In this paper some results on factorizations of the Fibonacci–like numbers U n and their squares are given. We find the factorizations using the circulant matrices, their determinants and eigenvalues.

Rational values of the arccosine function

Juan Varona (2006)

Open Mathematics

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We give a short proof to characterize the cases when arccos(√r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind.