On kernel polynomials and self-perturbation of orthogonal polynomials (Q1420357)

Let \((Q_n)_{n=0}^\infty\) be an orthogonal polynomial sequence with respect to the linear functional \(\sigma\) and let \((\alpha_n)_{n=0}^\infty\) a non identically zero sequence of complex numbers. Let the new polynomial sequence \((P_n)_{n=0}^\infty\) be defined by \[ P_n(x)=Q_n(x)-\alpha_n Q_{n-t}(x),\quad n,t\neq1,\quad \alpha_t\neq 0,\quad P_0(x)=Q_0(x)=1. \] In the present paper the authors study the conditions under which the sequence \((P_n)_{n=0}^\infty\) is also an orthogonal polynomial sequence. The case \(t=1\) was solved by Ya. L. Geronimus in 1940 and it was established that \(Q_n\) should be the so-called kernel polynomials \(K_n(x,\lambda)=\sum_{k=0}{P_k(x)P_k(\lambda)/{\langle \sigma,P_k^2\rangle}}\). Here the authors present a different proof easily extensible to the case \(t=2\) and establish the semiclassical character and the location of zeros of \(P_n\). For the case \(t=2\) they obtain several necessary and sufficient conditions for the sequence \((P_n)_{n=0}^\infty\) to be orthogonal. The case \(t=3\) does not lead to any orthogonal polynomial sequence as they also show in the paper.