부산대학교 도서관

Summary: ``Let $\bold u$ be a quasi-definite linear functional definedon the linear space of polynomials $\Bbb P$ with complex coefficients.For such a functional we can define a sequence of monic orthogonalpolynomials (SMOP in short) $(P_n)_{n\in\Bbb N}$, which satisfies athree term recurrence relation. If we increase the indices of therecurrence relation by one unity, then we get the sequence ofassociated polynomials of the first kind as solution. These polynomialswill be orthogonal with respect to a linear functional denoted by$\bold u^{(1)}$. In the literature two special transformations of thefunctional $\bold u$ are studied, the canonical Christoffeltransformation $\tilde{\bold u} = (x - c)\bold u$ and the canonicalGeronimus transformation $\hat{\bold u} = (x - c)^{-1}{\bold u} +M\delta_c$, where $c$ is a fixed complex number, $M$ is a free complexparameter and $\delta_c$ is the linear functional defined on $\Bbb P$as $\langle\delta_c, p(x) \rangle= p(c)$. For the Christoffeltransformation with SMOP $(\tilde P_n)_{n\in\Bbb N}$, we are interestedin analyzing the relation between the linear functionals $\boldu^{(1)}$ and $\bold u^{(1)}$. The super index denotes the linearfunctionals associated with the orthogonal polynomial sequences of thefirst kind $(P^{(1)}_n)_{n\in\Bbb N}$ and $(\tilde P^{(1)}_n)_{n\in\Bbb N}$, respectively. This problem is also studied forGeronimus transformations. Here we give close relations between theircorresponding monic Jacobi matrices by using the LU and ULfactorizations. To get this result, we first need to study the relationbetween $\bold u^{-1}$ (the inverse functional) and $\bold u^{(1)}$which can be given from a quadratic Geronimus transformation.''