2021-07-29T05:28:27Zhttps://teapot.lib.ocha.ac.jp/oaioai:teapot.lib.ocha.ac.jp:000346222021-03-01T22:19:14ZPell Equation. II. Mathematical structure of the family of the solutions of the Pell equationHosoya, Haruo400application/pdf紀要論文Mathematical structure of the families of solutions of Pell equations x^2-Dy^2=1 (called Pell-1) and x^2-Dy^2=-1 (Llep-1) are studied by using Cayley-Hamilton theorem. Besides discovery of several new recursive relations, it was found that the solutions (x_n, y_n) of Pell-1 are expressed by the Chebyshev polynomials of the first and second kinds, T_n and U_n, in terms of the smallest solutions (x_1, y_1). The solutions (t_n, u_n) of Pellep-1 which are the combination of Pell-1 and Llep-1 are expressed by using the conjugate Chebyshev polynomials. Similar results are obtained for the solutions of Pellep-4 through the modified Chebyshev polynomials and their conjugates. The solutions of Pellep-4 with several D values are found to form various interesting mathematical series of numbers, such as Fibonacci, Lucas, Pell numbers.お茶の水女子大学2007-01engdepartmental bulletin paperhttp://hdl.handle.net/10083/2403https://teapot.lib.ocha.ac.jp/records/34622AN0003395800298190お茶の水女子大學自然科學報告5721933https://teapot.lib.ocha.ac.jp/record/34622/files/KJ00004830817.pdfapplication/pdf673.9 kB2018-04-19