A generalized notion of input-to-state l(2)-gain is proposed for discrete-time switched linear control systems (SLCSs). Dependent on a discount factor of subsystem matrices, this generalized l(2)-gain provides new insight into input-to-state behaviors of the SLCSs under parameter variations. After establishing analytical properties of the generalized l(2)-gain, this paper focuses on the generating function approach to the study of the generalized l(2)-gain. Important properties of generating functions are derived, and it is shown that their radii of convergence characterize the generalized l(2)-gain. Furthermore, iterative algorithms are developed for computing the generating functions with proven uniform or exponential convergence. Numerical results show that these algorithms yield efficient estimates of both the generalized and classical l(2)-gains.