A structure relation for some specific orthogonal polynomials

By characterizing all orthogonal polynomials sequences $(P_n)_{n\geq 0}$ for which $(ax+b)(\triangle +2\,\mathrm{I})P_n(x(s-1/2))=(a_n x+b_n)P_n(x)+c_n P_{n-1}(x),\quad n=0,1,2,\dots,$ where $\,\mathrm{I}$ is the identity operator, $x$ defines a $q$-quadratic lattice, $\triangle f(s)=f(s+1)-f(s)$, and $(a_n)_{n\geq0}$, $(b_n)_{n\geq0}$ and $(c_n)_{n\geq0}$ are sequences of complex numbers, we derive some new structure relations for some specific families of orthogonal polynomials.

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