It has long been recognized that Fibonacci-type recurrence relations can be used to define a set of versatile polynomials \(\{ p_{n} (z)\}\) that have Fibonacci numbers and Chebyshev polynomials as special cases. We show that a tridiagonal matrix, which can be factored into the product \(AB\) of two special matrices \(A\) and \(B\), is associated with these polynomials. We apply tools that have been developed to study the supersymmetry of Hamiltonians that have a tridiagonal matrix representation in a basis to derive a set of partner polynomials \(\{ p_{n}^{( + )} (z)\}\) associated with the matrix product \(BA\). We find that special cases of these polynomials share similar properties with the Fibonacci numbers and Chebyshev polynomials. As a result, we find two new sum rules that involve the Fibonacci numbers and their product with Chebyshev polynomials.

早已认识到，斐波那契类型的递归关系可用于定义一组通用的多项式\（\ {p_ {n}（z）\} \），这些多项式具有斐波那契数和切比雪夫多项式作为特例。我们证明了可以将两个特殊矩阵\（A \）和\（B \）的乘积\（AB \）分解为一个三对角矩阵与这些多项式相关联。我们应用已开发的工具来研究具有三对角矩阵表示形式的哈密顿量的超对称性，以得出一组伙伴多项式\（\ {p_ {n} ^ {（+）}（z）\} \）与矩阵乘积\（BA \）相关联。我们发现这些多项式的特殊情况与斐波那契数和切比雪夫多项式具有相似的性质。结果，我们发现了两个新的和规则，涉及斐波那契数及其与切比雪夫多项式的乘积。