A characterization of Meixner orthogonal polynomials via a certain transfert operator

Here we consider a certain transfert operator $\mathrm{M}_{(c,\omega)}=I_{\mathcal{P}}-c \, \tau_{\omega},$ $\omega\neq0,$ ${c \in \mathbb{R}-\{0,1\},}$ and we prove the following statement: up to an affine transformation, the only orthogonal sequence that remains orthogonal after application of this transfert operator is the Meixner polynomials of the first kind.