On associated polynomials and decay rates for birth--death processes.
Consider a sequence of monic polynomials \(\{P_n(x): n= 0,1,\dots\}\) satisfying the familiar recurrence relation \[ \begin{gathered} P_n(x):= (x- c_n)(P_{n-1}(x)- d_n P_{n-2}(x),\quad n> 1,\\ P_0(x):= 1,\quad P_1(x):= x- c_1,\end{gathered} \] where \(c_n\) is real and \(d_n> 0\). By a theorem of Favard, there exists a positive Borel measure \(\psi\) of total mass 1 on the real axis, with respect to which the polynomials \(\{P_n(x)\}\) are orthogonal, that is, \[ \int^\infty_{-\infty} P_n(x) P_m(x)\psi(dx)= k_n\delta_{n,m},\quad n,m\geq 0, \] where \(K_n> 0\). Given the sequences \(\{c_n: n= 1,2,\dots\}\) and \(\{d_n: n= 1,2,\dots\}\), the corresponding sequence of associated polynomials \(\{\widetilde P_n(x): n=0,1,\dots\}\) is defined by the recurrence of the above type in which \(c_n\) and \(d_n\) are replaced by \(c_{n+1}\) and \(d_{n+1}\), respectively. Clearly, the associated polynomials \(\{\widetilde P_n(x)\}\) are also orthogonal with respect to another Borel measure of total mass 1 on the real axis. Denote this associated measure by \(\widetilde\psi\). The main aim of the present paper is to obtain information about the measure \(\psi\) from (partial) knowledge of the measure \(\widetilde\psi\). This analysis is motivated by applications in the setting of birth-death processes.
scientific article; zbMATH DE number 1915723
scientific article