I am currently studying difference equations and came across a problem where I have a solution to a homogeneous difference equation in the form:
$$y_n = (c_1 + c_2n + c_3n^3)7^n,$$
where $c_1, c_2,$ and $c_3$ are constants, and $n$ is the time index. The task is to find the original difference equation that this solution satisfies.
So we can separate the general solution into three solutions: $$\alpha_1 = 7^n$$ $$\alpha_2 = n7^n$$ $$\alpha_3 = n^27^n$$
In order to do so, I attempted to express it as a system of equations:
$$ A = \begin{pmatrix} 7^2 & 7 & 1 \\ 7^2(n+2) & 7(n+1) & n \\ 7^2(n+2) & 7(n+1)^2 & n^2 \end{pmatrix}, \\ b = \begin{pmatrix} 7^3 \\ 7^3(n+3) \\ 7^3(n+3)^2 \end{pmatrix}, $$
$$
and tried to solve for $Ax = b$.
However, I am struggling to make progress with this approach. Is there a more efficient or standard method for finding the difference equation corresponding to a given solution? Any hints or references would be greatly appreciated.
Analogously to a linear ODE, the difference operator ($\Delta$) should appear as
$$ \left(\Delta-r\right)^3y_n = 0\ \ \ (\text{a triple root}) $$
or
$$ \left(\Delta^3-3r\Delta^2+3r^2\Delta-r^3\right)y_n=0 $$
here $r = 7$ hence
$$ \left(\Delta^3-21\Delta^2+147\Delta-343\right)y_n=0 $$