A problem of Diophantos-Fermat and Chebyshev polynomials of the second kind. (English) Zbl 0844.11013

Udrea, Gheorghe

Port. Math. 52, No. 3, 301-304 (1995).

Chebyshev polynomials \((U_n )_{n\geq 0}\) are defined by the recurrence relation \(U_{n+1} (x)= 2xU_n (x)- U_{n-1} (x)\) where \(U_0 (x)=1\) and \(U_1 (x)= 2x\). The following theorem is proved for these polynomials: If \((U_n)\) is the sequence of Chebyshev polynomials of the second kind, then the product of any two distinct elements of the set \(\{U_n, U_{m+2r}, U_{m+4r}, 4U_{m+r} U_{m+2r} U_{m+3r}\}\), increased by \(U^2_a U^2_b\) for suitable positive integers \(a\) and \(b\), is a perfect square for any natural numbers \(m\), \(r\) and complex number \(x\). From the proof of the theorem some identities follow for Fibonacci numbers.

Reviewer: Péter Kiss (Eger)

MSC:

11B39

Fibonacci and Lucas numbers and polynomials and generalizations

Keywords:

recurrence relation; Chebyshev polynomials of the second kind; perfect square; Fibonacci numbers