Random Walks determined by a class of generalized Fibonacci polynomials

In this work, we briefly review the definitions of random walks and birth-and-death processes, as
these topics are well established. For further details, the interested reader is referred to Dominguez
[1] and references therein. These stochastic processes constitute a special class of Markov processes
with a discrete state space. We derive the one-step transition matrices for these two types of Markov
processes in discrete time. With these matrices at hand, we aim to compute their corresponding
n-step transition probabilities, which are given by the entries of the n–th power of the one–step
transition matrix. A successful method for this computation is the Karlin-McGregor representation;
see Karlin and McGregor [4] for a discussion on the relationship between orthogonal polynomials
and random walks. Our interest in studying orthogonal polynomials lies in understanding which
random walks are induced by such polynomials (see, for instance, [2]). In this work, we are
particularly interested in determining which generalized Fibonacci polynomials (see, for instance,
[3]) induce or characterize a random walk.