Hahn's problem with respect to some perturbations of the raising operator $(X-c)$

In this paper, we study the Hahn's problem with respect to some raising operators perturbed of the operator $X-c$, where $c$ is an arbitrary complex number. More precisely, the two following characterizations hold: up to a normalization, the $q$-Hermite (resp. Charlier) polynomial is the only $H_{\alpha,q}$-classical (resp. \linebreak $\mathcal{S}_{\lambda}$-classical) orthogonal polynomial, where $H_{\alpha, q}:=X+\alpha H_q$ and $\mathcal{S}_{\lambda}:=(X+1)-\lambda\tau_{-1}$.