Combinatorial Trigonometry with Chebyshev Polynomials

The Chebyshev polynomials have many beautiful properties and countless applications, arising in a variety of continuous settings. They are a sequence of orthogonal polynomials appearing in approximation theory, numerical integration, and differential equations. In this paper we approach them instead as discrete objects, counting the sum of weighted tilings. Using this combinatorial approach, one can prove numerous identities, as is done in [2, 3, 7]. In this note we provide a combinatorial proof of perhaps the most fundamental of Chebyshev properties, namely the trigonometric identity cos(nθ) = Tn(cos θ), where Tn is the Chebyshev polynomial of the first kind. We also provide combinatorial proofs of other trigonometric identities, including those involving Chebyshev polynomials of the second kind. The Chebyshev polynomials of the first kind are defined by T0(x) = 1, T1(x) = x, and for n ≥ 2, Tn(x) = 2xTn−1(x)− Tn−2(x). The next few polynomials are T2(x) = 2x −1, T3(x) = 4x−3x, T4(x) = 8x−8x+1, T5(x) = 16x 5 − 20x + 5x. The Chebyshev polynomials of the second kind differ only in the initial conditions. They are defined by U0(x) = 1, U1(x) = 2x, and for n ≥ 2, Un(x) = 2xUn−1(x)− Un−2(x). The next few polynomials are U2(x) = 4x 2 − 1, U3(x) = 8x − 4x, U4(x) = 16x − 12x + 1, U5(x) = 32x 5 − 32x + 6x.