标题： 两个广义斐波那契和卢卡斯多项式类的新结果及其在某些根式的化简中的应用 显示英文标题

标题： Novel Results of Two Generalized Classes of Fibonacci and Lucas Polynomials and Their Uses in the Reduction of Some Radicals

摘要： 本文关注于开发两种广义的斐波那契和卢卡斯多项式类之间的新的连接公式。 所有连接系数都涉及类型为$_2F_{1}(z)$的超几何函数，对于某些$z$。 一些著名的多项式如斐波那契、卢卡斯、佩尔、费马、佩尔-卢卡斯和费马-卢卡斯多项式之间的新的连接公式作为推导出的连接公式的特例被得出。 其中一些引入的公式推广了文献中的一些现有公式。 作为推导出的连接公式的两个应用，给出了一些著名数之间的新公式，并且也推导出了一些特定的带有权重的定积分的新闭合公式。 基于使用两种广义的斐波那契和卢卡斯多项式，开发了一些奇数和偶数根式的新的约简公式。 显示英文摘要

摘要： This paper is concerned with developing some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. All the connection coefficients involve hypergeometric functions of the type $_2F_{1}(z)$, for certain $z$. Several new connection formulae between some famous polynomials such as Fibonacci, Lucas, Pell, Fermat, Pell-Lucas, and Fermat-Lucas polynomials are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.