A Tiling Approach to Chebyshev Polynomials

The author grants HarveyMudd College the nonexclusive right to make this work available for noncommercial, educational purposes, provided that this copyright statement appears on the reproduced materials and notice is given that the copy-ing is by permission of the author. To disseminate otherwise or to republish re-quires written permission from the author. We present a combinatorial interpretation of Chebyshev polynomials. The nth Chebyshev polynomial of the first kind, Tn(x), counts the sum of all weights of n-tilings using light and dark squares of weight x and dominoes of weight −1, and the first tile, if a square must be light. If we relax the condition that the first squaremust be light, the sum of all weights is the nth Chebyshev polynomial of the second kind, Un(x). In this paper we prove many of the beautiful Chebyshev identities using the tiling interpretation. Content