An inverse problem related to 2-orthogonal polynomials

Source abstract

Abstract This paper is interested in the analysis of the 2-orthogonality of a monic polynomial sequence { Q n } n ≥ 0 ${\left\{{Q}_{n}\right\}}_{n\ge 0}$ constructed as a linear combination of a sequence of monic 2-orthogonal polynomials { P n } n ≥ 0 ${\left\{{P}_{n}\right\}}_{n\ge 0}$ with P n ( x ) + s n P n − 1 ( x ) = Q n ( x ) + t n Q n − 1 ( x ) $${P}_{n}\left(x\right)+{s}_{n}{P}_{n-1}\left(x\right)={Q}_{n}\left(x\right)+{t}_{n}{Q}_{n-1}\left(x\right)$$ where s n , t n ∈ C ${s}_{n},{t}_{n}\in \mathbb{C}$ , s 1 ≠ t 1 and s n t n ≠ 0 for all n ≥ 1. Also, we provide the connection between the truncated banded Hessenberg matrices associated with { P n } n ≥ 0 ${\left\{{P}_{n}\right\}}_{n\ge 0}$ and { Q n } n ≥ 0 ${\left\{{Q}_{n}\right\}}_{n\ge 0}$ . Moreover, the applicability of the obtained result is illustrated by an example.