Applications of Waring's formula to some identities of Chebyshev polynomials

Derivations and identities for Chebyshev polynomials of the first and second kinds

In this paper we follow the general approach, proposed earlier by the first author, which is derived from the invariant theory field and provides a way of obtaining of the polynomial identities for any arbitrary polynomial family. We introduce the notion of Chebyshev derivations of the first and second kinds, which is based on the polynomial algebra, and corresponding specific differential operators. We derive the elements of their kernels and prove that any element of the kernel of the derivations defines a polynomial identity satisfied by the Chebyshev polynomials of the first and second kin…

双和罗格斯-拉曼努金型恒等式

Double-sum Rogers-Ramanujan type identities

本文通过Chebyshev多项式展开式与正交关系，推导出双和罗格斯-拉曼努金型恒等式，扩展了安德鲁斯、史、孙和姚等人的已有结果。

新的二项式恒等式用于斐波那契、卢卡斯和广义斐波那契序列的多重索引

New Binomial Identities for Fibonacci, Lucas, and Generalized Fibonacci Sequences with Multiple Indices

本文提出新的恒等式，通过卢卡斯数的幂和二项式系数表示斐波那契、卢卡斯和广义斐波那契序列的多重索引项，利用对称多项式（瓦ring公式）应用于经典Binet公式。

Rogers-Ramanujan type identities and Chebyshev Polynomials of the third kind

It is known that $q$-orthogonal polynomials play an important role in the field of $q$-series and special functions. During studying Dyson's "favorite" identity of Rogers--Ramanujan type, Andrews pointed out that the classical orthogonal polynomials also have surprising applications in the world of $q$. By inserting Chebyshev polynomials of the third and the fourth kinds into Bailey pairs, Andrews derived a family of Rogers--Ramanujan type identities and also results related to mock theta functions and Hecke--type series. In this paper, by constructing a new Bailey pair involving Chebyshev pol…

Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle

Using Chebyshev polynomialsof both kinds, we construct rational fractions which are convergents of the smallest root of $x^2-αx+1$ for $α=3,4,5,\dots$.Some of the underlying identities suggest an identity involving binomialcoefficients which leads to a triangular array sharing many propertieswith Pascal's triangle.

切比雪夫、斐波那契与卢卡斯多项式之间通过三角和建立的关系

Relations between Chebyshev, Fibonacci and Lucas polynomials via trigonometric sums

本文通过有限三角和建立斐波那契、卢卡斯与切比雪夫多项式间的新恒等式，并给出其物理解释及与图有效电阻的联系。