Fibonacci, Chebyshev, and orthogonal polynomials. (English) Zbl 1138.11308

Aharonov, Dov; Beardon, Alan; Driver, Kathy

Am. Math. Mon. 112, No. 7, 612-630 (2005).

Summary: The Fibonacci numbers satisfy a second-order linear recurrence relation, and a variety of identities, and they have the property that the \(m\)th term divides the \(n\)th term precisely when \(m\) divides \(n\). These properties are also shared by some Chebyshev polynomials, some solutions of second-order linear recurrence relations with constant coefficients, and some solutions of some linear recurrence relations with variable coefficients. In this expository article, we attempt to explain why this is so.

MSC:

11B39	Fibonacci and Lucas numbers and polynomials and generalizations
01A35	History of mathematics in Late Antiquity and medieval Europe
01A45	History of mathematics in the 17th century
01A55	History of mathematics in the 19th century
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)