Application of generating functions to problems of random walk

We consider a problem on determining the first hit time of the positive semi-axis under a homogenous discrete integer random walk on a line. More precisely, the object of our study is the graph of the generating function of the mentioned random variable. For the random walk with the maximal positive increment $1$, we obtain the equation on the implicit generating function, which implies the rationality of the inverse generating function. In this case, we find the mathematical expectation and dispersion for the first hit time of a positive semi-axis under a homogenous discrete integer random walk on a line. We describe a general method for deriving systems of equations for the first hit time of a positive semi-axis under a homogenous discrete integer random walk on a line. For a random walk with increments $-1$, $0$, $1$, $2$ we derive an algebraic equation for the implicit generating function. We prove that a corresponding planar algebraic curve containing the graph of generating function is rational. We formulate and prove several general properties of the generating function the first hit time of the positive semi-axis under a homogenous discrete integer random walk on a line.