Let $Z_t$ be a birth-death process with birth rates $\lambda_n$ and death rates $\mu_n$ defined on the non-negative integers.
The family of orthogonal polynomials $\{P_n(x)\}_{n\ge0}$ associated with $Z_t$ is defined by the recurrence relation \begin{gather*} P_0(x)=1, \quad P_1(x)=\lambda_0+\mu_0-x\\ \lambda_n P_{n+1}(x)=(\lambda_n+\mu_n-x)P_n(x)-\mu_n P_{n-1}(x), \quad n>1 \end{gather*}
Now consider the infinitesimal generator matrix $Q_n$ of the process stopped at $n>2$. Its entries are given by \begin{align*} q_{i,j}= \begin{cases} \lambda_i &\quad\text{ if }j=i+1\\ -(\lambda_i+\mu_i) &\quad\text{ if }j=i\\ \mu_i &\quad\text{ if }j=i-1\\ 0 &\quad\text{ otherwise} \end{cases} \end{align*} for $0\le i,j\le n$. Finally, define the characteristic polynomial of $Q_n$ as $R_n(x)=\det(Q_n-x I_{n+1})$, where $I_m$ denotes the $m\times m$ identity matrix. Using the tridiagonal form of $Q_n$, such a determinant can be expanded as $$ R_n(x)=[-(\lambda_n+\mu_n)-x]R_{n-1}(x)-\mu_n\lambda_{n-1}R_{n-2}(x) $$ Assuming I have not made mistakes up to here, is there any relationship between the families of polynomials $\{P_n\}$ and $\{R_n\}$ (for some choice of $R_0(x)$ and $R_1(x)$), or simply between their zeros?
I am particularly interested in the linear case, $\lambda_n=n\lambda$ and $\mu_n=n\mu$. When $\lambda=\mu$, for example, $\{P_n\}$ are the associated Laguerre polynomials with index $1$. The polynomials $\{R_n\}$ look closely related, but I can't quite get the connection.
