It has been known for a long time that there is a close connection between stochastic processes and orthogonal polynomials. For example, N. Wiener 112] and K. Ito 56] knew that Hermite polynomials play an important role in the integration theory with respect to Brownian motion. In the 1950s D. G. Kendall 66], W. Ledermann and G. E. H. Reuter 67] 74], and S. Kar lin and J. L. McGregor 59] established another important connection. They expressed the transition probabilities of a birth and death process by means of a spectral representation, the so-called Karlin-McGregor representation, in...