A class of nonsymmetric orthogonal polynomials
Let \(\{P_ n(x)\}\) be a sequence of polynomials orthogonal with respect to a positive measure on [0,\(\infty)\), and let \(\{K_ n(x)\}\) denote the corresponding sequence of kernel polynomials with k-parameter 0. The authors study the sequence of polynomials \(\{R_ n(x)\}\) defined by \[ R_{2n}(x) = P_ n(x^ 2)\quad +\quad \vartheta_{2n}xK_{n-1}(x^ 2), \] \[ R_{2n+1}(x) = xK_ n(x^ 2)\quad +\quad \vartheta_{2n+1}P_ n(x^ 2) \] for some sequence \(\{\vartheta_ n\}\). A necessary and sufficient condition on \(\{\vartheta_ n\}\) is determined for \(\{R_ n(x)\}\) to be orthogonal with respect to a positive measure, and this measure is then expressed in terms of the measure with respect to which \(\{P_ n(x)\}\) is orthogonal. The paper concludes with a number of exmples of orthogonal polynomial sequences of this type.
scientific article; zbMATH DE number 4008824
scientific article