2020/01/21 by Moustafa Ibrahim, Ibrahim, Moustafa
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Black Holes and Theoretical Physics #Chaos-based Image/Signal Encryption #FOS: Mathematics #General Mathematics (math.GM) #math.GM
paper · pdf · doi:10.48550/arxiv.2001.08545
28 pages, 17 figures included. I revised it and made some changes to be more compact and clear. I welcome any comments, any cooperation for further research developments on the new notions given in the paper -like the notions of trajectories and orbits- and to answer the unsolved problems posted in this paper
openalex publication_date 2020/01/21 · arxiv created 2020/02/09 · arxiv updated 2020/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove new theorems for the polynomial expansions of xn ± yn in terms of the binary quadratic forms αx2 + βxy + αy2 and a x2 + bxy + a y2 . 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.