Orthogonal polynomials in the spectral analysis of Markov processes : birth death models and diffusion In pioneering work in the 1950s S. Karlin and J. McGregor showed that probabilistic aspects of certain Markov processes can be studied by analyzing the orthogonal eigenfunctions of associated operators. In the decades since many authors have extended and deepened this surprising connection between orthogonal polynomials and stochastic processes. This book gives a comprehensive analysis of the spectral representation of the most important one dimensional Markov processes namely discrete time birth death chains birth death processes and diffusion processes. It brings together all the main results from the extensive literature on the topic with detailed examples and applications. Also featuring an introduction to the basic theory of orthogonal polynomials and a selection of exercises at the end of each chapter it is suitable for graduate students with a solid background in stochastic processes as well as researchers in orthogonal polynomials and special functions who want to learn about applications of their work to probability Orthogonal polynomials Spectral representation of discrete time birth death chains Spectral representation of birth death processes Spectral representation of diffusion processes.