This paper is concerned with the problem of stability analysis of discrete time-delay systems. New finite sum inequalities, which encompass the ones based on Abel lemma or Wirtinger type inequality, are first proposed. The potential capability of the newly derived inequalities is then demonstrated by establishing less conservative stability conditions for some classes of linear discrete-time systems with delay. The derived stability criteria are theoretically and numerically proved to be less conservative than existing results. (C) 2016 Elsevier Ltd. All rights reserved.