Generalized Hahnʹs theorem

Let {Pn(x)}n=0∞ be an orthogonal polynomial system andL[·]=∑i=0k ai(x)Di D=ddxa linear differential operator of order k (⩾0) with polynomial coefficients. We find necessary and sufficient conditions for a polynomial sequence {Qn(x)}n=0∞ defined by Qn(x) ≔ L[Pn+r(r)(x)],n⩾0, to be also an orthogonal polynomial system. We also give a few applications of this result together with the complete analysis of the cases: (i) k=0,1,2 and r=0, and (ii) k=r=1.