A COMBINATION OF ORTHOGONAL POLYNOMIALS SEQUENCES: 2-5 TYPE RELATION

Abstract

In the present paper, a new characterization of the orthogonality of a monic polynomials sequence $\left\{ Q_{n}\right\} _{n\geq 0}$ is obtained. This is defined as a linear combination of another monic orthogonal polynomials sequence $\left\{ P_{n}\right\} _{n\geq 0}$ such as% \begin{equation*} Q_{n}(x)+r_{n}Q_{n-1}(x)=P_{n}(x)+s_{n}P_{n-1}(x)+t_{n}P_{n-2}\left( x\right) +v_{n}P_{n-3}\left( x\right) +w_{n}P_{n-4}(x),\ n\geq 0 \end{equation*}% where $w_{n}r_{n}\neq 0,$ for every $n\geq 5.$ Futhermore, the relation between the corresponding linear functionals is showed to be \begin{equation*} k\left( x-c\right) u=\left( x^{4}+ax^{3}+bx^{2}+dx+e\right) v \end{equation*}% where $c,$ $a,$ $b,$ $d,$ $e\in \mathbb{C}$ and $k\in \mathbb{C}\backslash\{0\}.$ Finally, an illustration using special case of the above type relation is given.