Complex equation simplification

Question

Let $k$ be a positive integer and $c_0$ a positive constant. Consider the following expression: \begin{equation} \left(2 i c_0-i+\sqrt{3}\right)^2 \left(-2 c_0+i \sqrt{3}+1\right)^k+\left(-2 i c_0+i+\sqrt{3}\right)^2 \left(-2 c_0-i \sqrt{3}+1\right)^k \end{equation} I would like to find a simple expression for the above in which only real numbers appear. It is clear that the above expression is always a real number since \begin{equation} \overline{\left(2 i c_0-i+\sqrt{3}\right)^2 \left(-2 c_0+i \sqrt{3}+1\right)^k}= \left(-2 i c_0+i+\sqrt{3}\right)^2 \left(-2 c_0-i \sqrt{3}+1\right)^k. \end{equation} But I am not able to simplify it. I am pretty sure I once saw how to do this in a complex analysis course but I cannot recall the necessary tools. Help is much appreciated.

Answer 1

i will outline one possible line of approach to this question. First, simplify a little. let the value of your expression be $E_k$ and let $$ a = \frac{1-2c_0}{\sqrt{3}} $$ then we may write $$ \frac{E_k}{3^{\frac{k}2 +1}} = (1-ai)^2(a+i)^k + (1+ai)^2(a-i)^k $$ using your remark concerning conjugacy we may write this as $$ \frac{E_k}{3^{\frac{k}2 +1}} = 2\mathfrak{Re}\bigg((1-ai)^2(a+i)^k \bigg) $$ now $$ \mathfrak{Re}\bigg((1-ai)^2(a+i)^k \bigg) = \mathfrak{Re}(1-ai)^2 \mathfrak{Re}(a+i)^k - \mathfrak{Im}(1-ai)^2 \mathfrak{Im}(a+i)^k \\ = (1-ai)^2 \mathfrak{Re}(a+i)^k - \mathfrak{Im}(1-ai)^2 \mathfrak{Im}(a+i)^k \\ (1-a^2)\mathfrak{Re}(a+i)^k +2a\mathfrak{Im}(a+i)^k $$ set $$ S_k = \mathfrak{Re}(a+i)^k \\ T_k = \mathfrak{Im}(a+i)^k $$ so we have the recurrence relations: $$ S_{k+1} = aS_k -T_k \\ T_{k+1} = S_k + a T_k $$ since $$ H_k =\mathfrak{Re}\bigg((1-ai)^2(a+i)^k \bigg) = (1-a^2)S_k +2aT_k $$ and $S_0 =1$, $T_0 =0$ we may write $$ H_k = \begin{pmatrix} 1-a^2 & 2a \end{pmatrix}\begin{pmatrix} a & -1 \\ 1 & a \end{pmatrix}^k\begin{pmatrix} 1 \\ 0 \end{pmatrix} $$ given the unreliability of my arithmetic, i shall venture no further, as this formulation may contain errors. however the method outlined may be useful as a procedure for calculating the value for a particular $k$.