Subsequences of Fibonacci and Lucas polynomials with geometric subscripts (Q2891256)

Let \(p\) and \(q\) be two real numbers, \(\Delta = p^2 - 4q \not= 0\) and NEWLINE\[NEWLINEU_0 = U_0(p, q) = 0, \;U_1 = U_1(p, q) = 1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEU_n \equiv U_n(p,q) = pU_{n-1}(p,q) - qU_{n-2}(p,q) \;\text{ for } n > 1,NEWLINE\]NEWLINE NEWLINE\[NEWLINEV_0 = V_0(p, q) = 2, \;V_1 = V_1(p, q) = p,NEWLINE\]NEWLINE NEWLINE\[NEWLINEV_n \equiv V_n(p,q) = pV_{n-1}(p,q) - qV_{n-2}(p,q) \;\text{ for } n > 1NEWLINE\]NEWLINE be bivariate Fibonacci and Lucas polynomials. Let NEWLINE\[NEWLINE\alpha = \frac{p + \sqrt{p^2 - 4q}}{2}, \;\beta = \frac{p - \sqrt{p^2 - 4q}}{2}.NEWLINE\]NEWLINE Obviously, \(U_n(1, -1)\) and \(V_n(1,-1)\) are the ordinary Fibonacci and Lucas numbers. A lot of properties of \(U_n\) and \(V_n\) are studied, e.g., it is proved the validity of NEWLINE\[NEWLINEU_{mn} = U_n\sum_{0 \leq k < m/2} (-q^n)^k \left ( \begin{matrix} m-k-1 \\ k \end{matrix} \right ) V_n^{m-2k-1},NEWLINE\]NEWLINE NEWLINE\[NEWLINEV_{mn} = \sum_{0 \leq k \leq m/2} (-q^n)^k \frac{m}{m - k} \left ( \begin{matrix} m-k \\ k \end{matrix} \right ) V_n^{m-2k},NEWLINE\]NEWLINE NEWLINE\[NEWLINEU_{2mn} = U_n\sum_{k=1}^m (-q^n)^{m-k} \left ( \begin{matrix} m+k-1 \\ 2k - 1 \end{matrix} \right ) V_n^{2k-1},NEWLINE\]NEWLINE NEWLINE\[NEWLINEU_{2mn+n} = U_n\sum_{k=0}^m (-q^n)^{m-k} \left ( \begin{matrix} m+k \\ 2k \end{matrix} \right ) V_n^{2k},NEWLINE\]NEWLINE NEWLINE\[NEWLINEV_{2mn} = \sum_{k=0}^m (-q^n)^{m-k} \frac{2m}{m + k} \left (\begin{matrix} m+k \\ 2k \end{matrix} \right ) V_n^{2k},NEWLINE\]NEWLINE NEWLINE\[NEWLINEV_{2mn+n} = \sum_{k=0}^m (-q^n)^{m-k} \frac{1+2m}{1 + m + k} \left (\begin{matrix} 1+m+k \\ 1+2k \end{matrix} \right ) V_n^{1+2k}.NEWLINE\]NEWLINE Also, for even \(k\), NEWLINE\[NEWLINEV_{k^{n+1}} = V_{k^n}^k - \sum_{i=0}^{k/2-1} \Delta^i U^{2i}_{k^n}q^{k^n(k/2-i)} \left \{ 2^{k-2i} \left ( \begin{matrix} \frac{k}{2} \\ i \end{matrix} \right ) - \frac{2k}{k + 2i} \left ( \begin{matrix} \frac{k}{2}+i \\ 2i \end{matrix} \right ) \right \},NEWLINE\]NEWLINE NEWLINE\[NEWLINEV_{k^{n+1}} = \Delta^{k/2} U_{k^n}^k - \sum_{i=0}^{k/2-1} (-q^{k^n})^{k/2-i} V_{k^n}^{2i} \left \{ 2^{k-2i} \left ( \begin{matrix} \frac{k}{2} \\ i \end{matrix} \right ) - \frac{2k}{k + 2i} \left ( \begin{matrix} \frac{k}{2}+i \\ 2i \end{matrix} \right ) \right \},NEWLINE\]NEWLINE and for odd \(k\), NEWLINE\[NEWLINEV_{k^{n+1}} = V_{k^n}^k - V_{k^n} \sum_{i=0}^{(k-3)/2} \Delta^i U^{2i}_{k^n}q^{k^n((k-1)/2-i} \left \{ 2^{k-2i-1} \left ( \begin{matrix} \frac{k-1}{2} \\ i \end{matrix} \right ) - \left ( \begin{matrix} \frac{k-1}{2}+i \\ 2i \end{matrix} \right ) \right \},NEWLINE\]NEWLINE NEWLINE\[NEWLINEU_{k^{n+1}} = \Delta^{(k-1)/2} U_{k^n}^k - U_{k^n} \sum_{i=0}^{(k-3)/2} (-q^{k^n})^{(k-1)/2-i} V^{2i}_{k^n} \left \{ 2^{k-2i-1} \left ( \begin{matrix} \frac{k-1}{2} \\ i \end{matrix} \right ) - \left ( \begin{matrix} \frac{k-1}{2}+i \\ 2i \end{matrix} \right ) \right \}.NEWLINE\]