A GENERALIZED SIMPLE RANDOM-WALK IN ONE-DIMENSION RELATED TO THE GAUSSIAN POLYNOMIALS

A generalization of the relation between the simple random walk on a r egular lattice and the diffusion equation in a continuous space is des cribed. In one dimension we consider a random walk of a walker with ex ponentially decreasing mobility with respect to time. It has an exact solution of the conditional probability, that is expressed in terms of the Gaussian polynomials, a generalization of binomial coefficients. Taking a suitable continuum limit we obtain the corresponding transpor t equation from the recursion relation of the discrete random walk pro cess. The kernel of this differential equation is also directly obtain ed from that conditional probability by the same continuum limit.