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

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
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