Abstract:

Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula

$$\frac{1}{1-ax-x^2}\ast \frac{1}{1-bx-x^2} = \frac{1-x^2}{1-abx-(2+a^2+b^2)x^2 -abx^3+x^4},$$

where $*$ denotes the Hadamard product. In a similar way, by considering tilings of a $2\times n$ rectangle with $1\times1$ and $1\times 2$ bricks in the top row, and $1\times1$ and $1\times n$ bricks in the bottom row, we find an explicit formula for the Hadamard product

$$\frac{1}{1-ax-x^2}\ast \frac{x^m}{1-bx-x^n}.$$