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A formula for the generating functions of powers of Horadam’s sequence. (English) Zbl 1053.05008

Let the sequence \(w_n=w_n(a,b;p,q)\) be given by the linear recurrence \(w_n=pw_{n-1}+ qw_{n-2}\) with \(w_0=a\) and \(w_1=b\). Horadam’s sequence is then defined by \[ {\mathcal H}_k(x)={\mathcal H}_k (x;a,b;p,q)= \sum_{n\geq 0} w^k_nx^n. \] The main result of this paper is
Theorem 1.1. The generating function for \({\mathcal H}_k(x)\) is give by \(\frac{\det(\delta_k)} {\det(\Delta_k)}\), where
\[ \begin{aligned} \Delta_k&= \left( \begin{matrix} 1-p^kx-q^kx^2 & -xp^{k-1}q^1 {k\choose 1} & \dots & -xp^2 q^{k-2} {k\choose k-2} & -xpq^{k-1} {k\choose k-1}\\ -p^{k-1}x & 1-xp^{k-2}q^k {k-1\choose 1} & \dots & -xpq^{k-2} {k-1\choose k-2} & -xq^{k-1} {k-1\choose k-1}\\ \vdots & \vdots & \vdots & \vdots & \vdots \\ -p^2x & -xpq^{1{2\choose 1}} & \dots & 1 & 0\\ -px & -xq^1{1 \choose 1} & \dots & 0 & 1 \end{matrix}\right),\\ \delta_k&= \left( \begin{matrix} a^kx+g_kx & -xp^{k-1} q^1{k\choose 1} & \dots & -xp^2q^{k-2} {k\choose k-2} & -xpq^{k-1} {k\choose k-1}\\ g_{k-1}x & 1-xp^{k-2}q^1{k-1 \choose 1} & \dots & -xpq^{k-2} {k-1\choose k-2} & -xq^{k-1} {k-1\choose k-1}\\ -p^{k-1}x & -xp^{k-3}q^1 {k-2\choose 1} & \dots & -xq^{k-2} {k-2\choose k-2} & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots\\ g_2x & -xpq^1 {2\choose 1} & \dots & 1 & 0\\ g_1x & -xq^1 {1\choose 1} & \dots & 0 & 1\end{matrix}\right) \end{aligned} \]
and \(g_j=(b^j- a^jp^j)a^{k-j}\), for \(j=1,\dots, k\).

MSC:

05A15 Exact enumeration problems, generating functions
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11B37 Recurrences
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Online Encyclopedia of Integer Sequences:

Fourth powers of Lucas numbers A000032.
a(n) = Pell(n)^3.