On a new combination of orthogonal polynomials sequences

In this paper, we are interested in the following inverse problem. We assume that {Pn}n≥0 is a monic orthogonal polynomials sequence with respect to a quasi-definite linear functional u and we analyze the existence of a sequence of orthogonal polynomials {Qn}n≥0 such that we have a following decomposition Qn(x)+rnQn-1(x)=Pn(x)+snPn-1(x)+tnPn-2(x)+vnPn-3(x), n≥0, when vnrn≠0, for every n≥4. Moreover, we show that the orthogonality of the sequence {Qn}n≥0 can be also characterized by the existence of sequences depending on the parameters rn, sn, tn, vn and the recurrence coefficients which remain constants. Furthermore, we show that the relation between the corresponding linear functionals is k(x-c)u=(x3+ax2+bx+d)v, where c,a,b,d∈C and k∈C∖{0}. We also study some subcases in which the parameters rn, sn, tn and vn can be computed more easily. We end by giving an illustration for a special example of the above type relation.