Associated orthogonal polynomials of the first kind and Darboux transformations

Abstract

Let $\un,$ be a quasi-definite linear functional defined on the space of polynomials $\mathbb{P}.$ For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) $(P_n)_{n\geq 0},$ which satisfies a three term recurrence relation. Shifting one unity the recurrence coefficient indices given the sequence of associated polynomials of the first kind which are orthogonal with respect to a linear functional denoted by $\un^{(1)}$. In the literature two special transformations of the functional $\un$ are studied, the canonical Christoffel transformation $\widetilde \un=(x-c)\un$ and the canonical Geronimus transformation $\widehat \un=\frac{\un}{(x-c)}+M\deltan_c$ , where $c$ is a fixed complex number, $M $ is a free parameter and $\deltan_c$ is the linear functional defined on $\mathbb{P}$ as $\prodint{\deltan_c,p(x)}=p(c).$ For the Christoffel transformation with SMOP $(\widetilde P_n)_{n\geq 0}$, we are interested in analyzing the relation between the linear functionals $ \un^{(1)}$ and $\widetilde{\un}^{(1)}.$ There, the super index denotes the linear functionals associated with the orthogonal polynomial sequences of the first kind $(P_n^{(1)})_{n\geq 0}$ and $(\widetilde P_n^{(1)})_{n\geq 0},$ respectively. This problem is also studied for Geronimus transformations. Here we give close relations between their corresponding monic Jacobi matrices by using the LU and UL factorizations. To get this result, we first need to study the relation between $\un^{-1}$ (the inverse functional) and $\un^{(1)}$ which can be expressed from a quadratic Geronimus transformation.