论文标题
$ x^n \ pm y^n $在二次形式方面的新扩展
New expansions for $x^n \pm y^n$ in terms of quadratic forms
论文作者
论文摘要
我们证明了$ x^n \ pm y^n $的多项式扩展的新定理，就二进制二进制形式而言$αx^2 +βxy +αy +αy^2 $和$ a x^2 + bxy + bxy + a y^2 $。本文提供了新的算术差分方法来计算系数。同样，本文在数字理论史上对众所周知的多项式身份进行了概括。本文凸显了新的多项式的出现，该序列统一了许多众所周知的序列，包括第一类和第二类的Chebyshev多项式，第一类和第二类的狄克森多项式，卢卡斯和斐波那契数字，梅尔森数字，梅尔森数字，佩尔·多项式，pell-lucas，pell-lucas polynomials和fynomials and fermats and fmermat。同样，本文强调了某些通过许多众所周知的多项式和序列的轨迹和轨道的概念的出现。 Lucas-fibonacci轨迹，Lucas-Pell轨迹，fibonacci-pell轨迹，佛比诺那acci-lucas轨迹，第一类的Chebyshev-Dickson轨迹，第二种的Chebyshev-Dickson轨迹，其他的轨迹以及本文中包括了新的轨迹。此外，卢卡斯轨道，斐波那契轨道，梅森尼轨道，卢卡斯 - 菲比尼奇轨道，费马特轨道以及其他是本文中包括的新轨道。
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.
