This chapter gives an introduction to orthogonal polynomials. It also includes the concept of the Stieltjes transform and some of its properties, which will play a very important role in the spectral analysis of discrete-time birth–death chains and birth–death processes. A section on the spectral theorem for orthogonal polynomials (or Favard’s theorem) will give insights into the relationship between tridiagonal Jacobi matrices and spectral probability measures. The chapter then focuses then on the classical families of orthogonal polynomials, of both continuous and discrete variables. These families are characterized as eigenfunctions of second-order differentials or difference operators of hypergeometric type solving certain Sturm–Liouville problems. These classical families are part of the so-called Askey scheme.

This chapter is devoted to the spectral analysis of discrete-time birth–death chains on nonnegative integers, which are the most basic and important discrete-time Markov chains. These chains are characterized by a tridiagonal one-step transition probability matrix. The so-called Karlin–McGregor integral representation formula of the n-step transition probability matrix is obtained in terms of orthogonal polynomials with respect to a probability measure with support inside the interval [–1, 1]. An extensive collection of examples related to orthogonal polynomials is provided, including gambler’s ruin, the Ehrenfest model, the Bernoulli–Laplace model and the Jacobi urn model. The chapter concludes with applications of the Karlin–McGregor formula to probabilistic aspects of discrete-time birth–death chains, such as recurrence, absorption, the strong ratio limit property and the limiting conditional distribution. Finally, spectral methods are applied to discrete-time birth–death chains on the integers, which are not so much studied in the literature.

This chapter is devoted to the spectral analysis of one-dimensional diffusion processes, which are the most basic and important continuous-time Markov processes where now the state space is a continuous interval contained in the real line. Diffusion processes are characterized by an infinitesimal operator which is a second-order differential operator with drift and diffusion coefficients. A spectral representation of the transition probability density of the process is obtained in terms of the orthogonal eigenfunctions of the corresponding infinitesimal operator, for which a Sturm–Liouville problem with certain boundary conditions will be solved. An analysis of the behavior of these boundary points will also be made. An extensive collection of examples related to special functions and orthogonal polynomials is provided, including the Brownian motion with drift and scaling, the Orstein–Uhlenbeck process, a population growth model, the Wright–Fisher model, the Jacobi diffusion model and the Bessel process, among others. Finally, the concept of quasi-stationary distributions is studied, for which the spectral representation plays an important role.

This chapter is devoted to the spectral analysis of birth–death processes on nonnegative integers, which are the most basic and important continuous-time Markov chains. These processes will be characterized by an infinitesimal operator which is a tridiagonal matrix whose spectrum is always contained in the negative real line (including 0). The Karlin–McGregor integral representation formula of the transition probability functions of the process is obtained in terms of orthogonal polynomials with respect to a probability measure with support inside a positive real interval. Although many of the results are similar or equivalent to those of discrete-time birth–death chains, the methods and techniques are quite different. The chapter gives an extensive collection of examples related to orthogonal polynomials, including the M/M/k queue for any k servers, the continuous-time Ehrenfest and Bernoulli–Laplace urn models, a genetics model of Moran and linear birth–death processes. As in the case of discrete-time birth–death chains, the Karlin–McGregor formula is applied to the probabilistic aspects of birth–death processes, such as processes with killing, recurrence, absorption, the strong ratio limit property, the limiting conditional distribution, the decay parameter, quasi-stationary distributions and bilateral birth–death processes on the integers.