$x^n \pm y^n$的二次型新展开式
New expansions for $x^n \pm y^n$ in terms of quadratic forms
摘要：
证明了$x^n \pm y^n$关于二元二次型$αx^2 + βxy + αy^2 $和$a x^2 + bxy + a y^2 $的多项式展开式的新定理。本文提出了一种新的算术微分法来计算系数。对数论史上著名的多项式恒等式进行了推广。本文重点介绍了一类新的多项式的出现，统一了许多著名的序列，包括切比雪夫多项式的第一和第二类，迪克森多项式的第一和第二类，卢卡斯和斐波那契数，梅森数，佩尔多项式，佩尔-卢卡斯多项式，和费马数。此外，本文强调了出现的概念的轨迹和轨道的某些整数，通过许多著名的多项式和序列。Lucas-Fibonacci轨线、Lucas-Pell轨线、Fibonacci-Pell轨线、Fibonacci-Lucas轨线、第一类Chebyshev-Dickson轨线、第二类Chebyshev-Dickson轨线等都是本文所包括的新轨线。此外，卢卡斯轨道，斐波那契轨道，梅森轨道，卢卡斯-斐波那契轨道，费马轨道，和其他新的轨道包括在本文中。
英文摘要：
We prove new theorems for the polynomial expansions of $x^n \pm y^n$ in terms of the binary quadratic forms $αx^2 + βxy + αy^2 $ and $a x^2 + bxy + a y^2 $. The paper gives new arithmetic differential approach to compute the coefficients. Also, the paper gives generalization to well-known polynomial identity in the history of number theory. The paper highlights the emergence of a new class of polynomials that unify many well-known sequences including the Chebyshev polynomials of the first and second kind, Dickson polynomials of the first and second kind, Lucas and Fibonacci numbers, Mersenne numbers, Pell polynomials, Pell-Lucas polynomials, and Fermat numbers. Also, this paper highlights the emergence of the notions of trajectories and orbits of certain integers that passes through many well-known polynomials and sequences. The Lucas-Fibonacci trajectory, the Lucas-Pell trajectory, the Fibonacci-Pell trajectory, the Fibonacci-Lucas trajectory, the Chebyshev-Dickson trajectory of the first kind, the Chebyshev-Dickson trajectory of the second kind, and others are new trajectories included in this paper. Also, the Lucas orbit, Fibonacci orbit, Mersenne orbit, Lucas-Fibonacci orbit, Fermat orbit, and others are new orbits included in this paper.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
