Counting on Chebyshev Polynomials

Abstract

where Tn is the Chebyshev polynomial of the first kind, defined by T0(x) = 1, Tx(x) = x, and for n > 2, Tn(x) = 2xTn.x{x) - Tn_2(x). (2) For example, T2(x) = 2x2 - 1, T3(x) = 4x3 - 3x, T4(x) = 8x4 - 8x2 + 1. This gen erates the familiar trigonometric identity eos(20) = 2 cos2 9 ? 1, and the less familiar cos(36>) = 4 cos3 9 - 3 cos 9 and cos(4#) = 8 cos4 9 - 8 cos2 9 + 1. If we change the initial conditions to be Uo(x) = 1 and Ux(x) = 2x, but keep the same recurrence Un(x) = 2xUn-i(x) - Un-2(x), we get the Chebyshev polynomials of the second kind. For instance, U2(x) = Ax1 ? 1, U3(x) = Sx3 - Ax, U4(x) = 16x4 - I2x2 + 1. The Chebyshev polynomials generate many fundamental sequences, including the constant sequence, the sequence of integers, and the Fibonacci numbers. It's easy to show that for all n > 0, Tn(l) = 1 and Un(l) = n + l, Tn(~l) = (-If, Un(-l) = (? l)n(n + 1). When we substitute complex numbers, such as x = i/2, the Fibonacci and Lucas numbers appear. Specifically,