On a new combination of orthogonal polynomials sequences

In this paper, we are interested in the following inverse problem. We assume that $\{P_{n}\} _{n\geq 0}$ 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 $\{ Q_{n}\} _{n\geq 0}$ such that we have a following decomposition $Q_{n}(x)+r_{n}Q_{n-1}(x)=P_{n}(x)+s_{n}P_{n-1}(x)+t_{n}P_{n-2}(x) +v_{n}P_{n-3}( x)$, $n\geq 0$, when $v_{n}r_{n}\neq 0,$ for every $n\geq 4.$ Moreover, we show that the orthogonality of the sequence $\{Q_{n}\}_{n\geq 0}$ can be also characterized by the existence of sequences depending on the parameters $r_{n}$, $s_{n}$, $t_{n}$, $v_{n}$ and the recurrence coefficients which remain constants. Furthermore, we show that the relation between the corresponding linear functionals is $k( x-c) u=( x^{3}+ax^{2}+bx+d) v$, where $c, a, b, d\in \mathbb{C}$ and $k\in \mathbb{C}\setminus \{0\}$. We also study some subcases in which the parameters $r_{n},$ $s_{n},$ $t_{n}$ and $v_{n}$ can be computed more easily. We end by giving an illustration for a special example of the above type relation.