Higher Order Coherent Pairs

In this paper, we study necessary and sufficient conditions for the relation $$begin{array}{@{}l}P_n^{{[r]}}(x) + a_{n-1,r} P_{n-1}^{{[r]}}(x)= R_{n-r}(x) + b_{n-1,r} R_{n-r-1}(x),[5pt]quad a_{n-1,r}neq0, ngeq r+1,end{array}$$ where {P _n (x)} _n??0 and {R _n (x)} _n??0 are two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals $mathcal{U},mathcal{V}$ , respectively, or associated with two positive Borel measures ?? ₀,?? ₁ supported on the real line. We deduce the connection with Sobolev orthogonal polynomials, the relations between these functionals as well as their corresponding formal Stieltjes series. As sake of example, we find the coherent pairs when one of the linear functionals is classical.