Title: Zeros and Orthogonality of generalized Fibonacci polynomials Show Chinese title

Title: 广义斐波那契多项式的零点和正交性

Abstract: This paper analyzes the concept of orthogonality in second-order polynomial sequences that have Binet formula similar to that of the Fibonacci and Lucas numbers, referred to as Generalized Fibonacci Polynomials (GFP). We give a technique to find roots of the GFP. As a corollary of this result, we give an alternative proof of a special case of Favard's Theorem. The general case of Favard's Theorem guarantees that there is a measure to determine whether a sequence of second-order polynomials is orthogonal or not. However, the theorem does not provide an explicit such measure. Our special case gives both the explicit measure and the relationship between the second-order recurrence and orthogonality, demonstrating whether the GFP polynomials are orthogonal or not. This allows us to classify which of familiar GFPs are orthogonal and which are not. Some familiar orthogonal polynomials include the Fermat, Fermat-Lucas, both types of Chebyshev polynomials, both types of Morgan-Voyce polynomials, and Vieta and Vieta-Lucas polynomials. However, we prove that the Fibonacci, Lucas, Pell, and Pell-Lucas sequences are not orthogonal. In Section \ref{sectionrw}, we give a brief description of discrete--time and continuous--time Morkov chains with special emphasis on birth-and-death stochastic processes. We find sufficient conditions on the polynomial's coefficients under which a given family of orthogonal polynomial induces a Markov chain. These families of orthogonal polynomials include Chebyshev polynomials of first kind and Fermat-Lucas. In the final section, we highlight some connections between orthogonal polynomials and Markov processes. These relations are not new but seem to have been somewhat forgotten. We do so to draw the attention of researchers in the orthogonal polynomial and probability communities for further collaboration. Show Chinese abstract

Abstract: 本文分析了具有类似斐波那契数和卢卡斯数的Binet公式的二阶多项式序列中的正交性概念，称为广义斐波那契多项式（GFP）。 我们提供了一种找到GFP根的技术。 作为该结果的一个推论，我们给出了法瓦尔定理一个特例的另一种证明。 法瓦尔定理的一般情况保证存在一个测度来确定一个二阶多项式序列是否正交。 然而，该定理并未提供一个显式的这样的测度。 我们的特例给出了显式的测度以及二阶递推关系与正交性之间的关系，证明了GFP多项式是否正交。 这使我们能够分类哪些熟悉的GFP是正交的，哪些不是。 一些熟悉的正交多项式包括费马多项式、费马-卢卡斯多项式、两种类型的切比雪夫多项式、两种类型莫根-沃伊茨多项式以及维埃塔和维埃塔-卢卡斯多项式。 然而，我们证明了斐波那契、卢卡斯、佩尔和佩尔-卢卡斯序列不是正交的。 在第\ref{sectionrw}节中，我们简要描述了离散时间与连续时间马尔可夫链，特别强调出生-死亡随机过程。 我们找到了多项式系数的充分条件，在这些条件下，给定的正交多项式族会诱导一个马尔可夫链。 这些正交多项式族包括第一类切比雪夫多项式和费马-卢卡斯多项式。 在最后一节中，我们强调了正交多项式与马尔可夫过程之间的一些联系。 这些关系并不新鲜，但似乎已被部分遗忘。 我们这样做是为了引起正交多项式和概率领域研究人员的注意，以促进进一步的合作。