Hadamard Products and Tilings
Abstract
Louis W. Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula 1/(1-ax-x2) * 1/(1-bx-x2) = (1-x2)/(1-abx-(2+a2+b2)x2 -abx3+x4), where * denotes the Hadamard product. In a similar way, by considering tilings of a 2 by n rectangle with 1 by 1 and 1 by 2 bricks in the top row, and 1 by 1 and 1 by n bricks in the bottom row, we find an explicit formula for the Hadamard product 1/(1-ax-x2) * xm/(1-bx-xn).
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