Recently, the problem of analysis and optimal control of discrete time‐invariant systems has been extensively studied using finite series expansion of discrete orthogonal polynomials. This paper is to extend the applicable scope of discrete orthogonal polynomials to discrete time‐varying systems. The finite set of Hahn polynomials {qik)],i=0, 1, …,N} is chosen as the finite series expansion basis due to its general form and useful properties. First, for treating the product of two discrete‐time functions by Hahn series expansion, a new algorithm is derived to compute the Hahn series expansion coefficients of productsqi(k)qj(k),i, j=0, 1, …,N. These Hahn coefficients are then used to establish a product operational matrix for relating the Hahn coefficient vector of a product function to those of its component functions. This product operational matrix, along with the relations for connecting the Hahn coefficient vectors of a discrete functionx(k) and its time‐shiftedx(k+1), is finally applied to derive computational algorithms for solving the problems of analysis and optimal control of discrete time‐varying systems via finite Hahn series. Computed results are provided to illustrate the applicability of the proposed algorithms.