Eigenfunctions of discrete-time LTI systems Given a discrete-time signal x[n], how does one determine if it is an eigenfunction of a stable, discrete-time LTI system?

For example, consider


$5^nu[n]$
$2^nu[-n-1]$


Intuitively, second one seems to be an eigenfunction of an LTI system while first one does not. Since the second sequence is absolutely summable while first one is not, is this the criteria for deciding?
i think the only family of functions that are eigenfunctions to LTI systems the sole exponential.

for continuous-time, if it's LTI:

$$ y(t) = \int\limits_{-\infty}^{+\infty} h(u) \ x(t-u) \ du $$

the eigenfunction is $x(t) = A e^{st}$ and the output is

$$ y(t) = H(s) \ x(t) $$

where

$$ H(s) = \int\limits_{-\infty}^{+\infty} h(t) \ e^{-st} \ dt $$

for discrete-time LTI systems, it is similar

$$ y[n] = \sum\limits_{m=-\infty}^{+\infty} h[m] \ x[n-m] $$

the eigenfunction is $x[n] = A z^n$ and the output is

$$ y[n] = H(z) \ x[n] $$

where

$$ H(z) = \sum\limits_{n=-\infty}^{+\infty} h[n] \ z^{-n} $$

often $s=j\Omega$ or $z=e^{j\omega}$ but it wouldn't have to be that.