Random Walks and Mixtures of Gamma Distributions

We consider the Wiener-Hopf integral equation describing random walk on the semi-axis in the case when a distribution density of a random variable is a two-sided mixture of Gamma distributions. The structure of the distribution function of the first entrance in $x < 0$ and $x \ge 0$ is revealed. A simple construction of these functions and corresponding renewal function is offered in the case of symmetric distribution with one exponent. Results of the work are expanding the class of problems of the random walk, allowing an effective solution. The method of the nonlinear factorization equations is applied.