[Paper Review] Hadamard Products and Tilings

Jong‐Hyun Kim|arXiv (Cornell University)|Mar 3, 2009

graph theory and CDMA systems6 references3 citations

TL;DR

This paper presents a combinatorial approach using tiling models to derive explicit formulas for Hadamard products of rational generating functions, specifically for expressions of the form $\frac{1}{1-ax-x^2} \ast \frac{x^m}{1-bx-x^n}$. By modeling the Hadamard product as weighted tilings of a $2\times n$ rectangle and decomposing tilings into prime blocks, the authors derive closed-form rational generating functions, generalizing Shapiro's earlier result for Chebyshev polynomials and extending it to asymmetric tiling constraints.

ABSTRACT

Louis W. Shapiro gave a combinatorial proof of a bilinear generating function for Chebyshev polynomials equivalent to the formula 1/(1-ax-x^2) * 1/(1-bx-x^2) = (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 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-x^2) * x^m/(1-bx-x^n).

Motivation & Objective

• To extend Shapiro's combinatorial proof of a bilinear generating function for Chebyshev polynomials to asymmetric tiling settings.
• To derive an explicit rational generating function for the Hadamard product $\frac{1}{1-ax-x^2} \ast \frac{x^m}{1-bx-x^n}$ using tiling decompositions.
• To generalize the MacMahon operator and Hadamard product framework to rational functions with non-uniform tiling constraints.
• To recover and re-derive known results, such as Han's identity, from the new tiling-based formalism.

Proposed method

• Model the Hadamard product as pairs of weighted tilings of a $2\times n$ rectangle, where the top row uses $1\times1$ and $1\times2$ tiles weighted by $a$, and the bottom row uses $1\times1$ and $1\times n$ tiles weighted by $b$.
• Define prime blocks as irreducible tiling units that cannot be factored further, and compute their generating function $P_n(x)$ to represent the recurrence structure of the tiling system.
• Use the MacMahon operator $\Omega_{\geq}$ to extract non-negative degree terms from Laurent series, enabling the derivation of generating functions from tiling decompositions.
• Classify first blocks (initial tiling segments) and subsequent prime blocks separately, leading to a rational generating function expressed as $Q_{m,n}(x)/(1 - P_n(x))$.
• Apply an identity from Lemma 2.1 relating Fibonacci-like polynomials: $f_{m-1}(a)f_{n-1}(a) - f_m(a)f_{n-2}(a) = (-1)^{\min(m-1,n-1)} f_{|m-n+1|-1}(a)$, to simplify coefficients in the generating function.
• Use the unique factorization of tilings into sequences of prime blocks to derive the final rational form of the Hadamard product.

Experimental results

Research questions

• RQ1 How can the Hadamard product $\frac{1}{1-ax-x^2} \ast \frac{x^m}{1-bx-x^n}$ be explicitly computed using combinatorial tiling models?
• RQ2 What is the generating function for the weighted first block in a tiling system where the top row starts with a $1\times m$ brick and the bottom row uses $1\times n$ bricks?
• RQ3 How does the tiling decomposition into prime blocks and first blocks lead to a rational closed-form expression for the Hadamard product?
• RQ4 Can Han's identity involving $\Omega_{\geq}$ be recovered from this tiling-based formalism?
• RQ5 What is the structure of the generating function when $b=0$, reducing the bottom row to only $1\times1$ tiles?

Key findings

• The Hadamard product $\frac{1}{1-ax-x^2} \ast \frac{x^m}{1-bx-x^n}$ is given by $\frac{f_m(a)x^m + b f_{m-1}(a)x^{m+1} + (-1)^{\min(m-1,n-1)} f_{|m-n+1|-1}(a) x^{m+n}}{1 - f_{n-2}(a)x^n}$ for $m \geq 1$, $n \geq 2$, with the denominator corrected to $1 - f_{n-2}(a)x^n$.
• When $b=0$, the formula simplifies to $\sum_{k \geq 0} f_{m+nk}(a) x^{m+nk} = \frac{f_m(a)x^m + (-1)^{\min(m-1,n-1)} f_{|m-n+1|-1}(a) x^{m+n}}{1 - (f_n(a) + f_{n-2}(a))x^n + (-1)^n x^{2n}}$, providing a rational generating function for arithmetic progressions of Fibonacci-like coefficients.
• For $m = qn + r$ with $0 < r < n$, the Hadamard product $\frac{x^m}{1-ax-x^2} \ast \frac{1}{1-x^n}$ is $\frac{f_{n-r}(a)x^{(q+1)n} + (-1)^{n-r-1} f_{|r-1|-1}(a) x^{(q+2)n}}{1 - (f_n(a) + f_{n-2}(a))x^n + (-1)^n x^{2n}}$, with a separate case for $m = qn$.
• The generating function for the first block in the tiling model is rational and derived via a recurrence based on prime block decomposition, enabling the full Hadamard product to be expressed as a rational function.
• The method recovers Han’s identity $\Omega_{\geq} \frac{1}{(1-zx-zx^2)(1-y/x-y/x^2)} = \frac{1+z^2y}{(1-2z)(1-3zy-z^2y-zy^2)}$ from the derived formula for $1/(1-ax-x^2) \ast 1/(1-bx-x^3)$.
• The paper establishes a general framework where the Hadamard product of rational generating functions corresponds to a tiling system with first blocks and prime block sequences, yielding a rational generating function via $Q_{m,n}(x)/(1 - P_n(x))$.