Identities for like-powers of Lucas sequences from algebraic identities
Volume 149 / 2017
Abstract
Let $a$ and $b$ be integers with $b(a^2+4b) \ne 0$. Let $u_0 = 0$, $u_1 = 1$, and $u_n = a u_{n-1} + b u_{n-2}$ for $n \ge 2$. Let $v_0 = 2$, $v_1 = a$, and $v_n = a v_{n-1} + b v_{n-2}$ for $n \ge 2$. Using algebraic identities we will prove some results, including the following ones. For integers $n \ge 0$ and $k \ge 1$, \begin{align*} u_{n+3k}^2 &= (v_{2k} + (-b)^k) u_{n+2k}^2 - (-b)^k (v_{2k} + (-b)^k) u_{n+k}^2 + (-b)^{3k} u_n^2 \\%[4pt] v_{n+3k}^2 &= (v_{2k} + (-b)^k) v_{n+2k}^2 - (-b)^k (v_{2k} + (-b)^k) v_{n+k}^2 + (-b)^{3k} v_n^2 \\%[4pt] u_{n+4k}^3 &= (v_{3k} + (-b)^k v_k) u_{n+3k}^3 - (-b)^k (v_{4k} + (-b)^k v_{2k} + 2 (-b)^{2k}) u_{n+2k}^3 \\%[4pt] &\quad + (-b)^{3k} (v_{3k} + (-b)^k v_k) u_{n+k}^3 - (-b)^{6k} u_n^3 \\%[4pt] v_{n+4k}^3 &= (v_{3k} + (-b)^k v_k) v_{n+3k}^3 - (-b)^k (v_{4k} + (-b)^k v_{2k} + 2 (-b)^{2k}) v_{n+2k}^3 \\%[4pt] &\quad + (-b)^{3k} (v_{3k} + (-b)^k v_k) v_{n+k}^3 - (-b)^{6k} v_n^3 . \end{align*} These results generalize some results of Gould (1963), Zeitlin and Parker (1963), Bicknell (1972), and Prodinger (1997).
