An Elementary Theory of Indefinite Summation Using Integral Transforms

We present a novel approach to indefinite summation through integral transform methods. We first establish the Laplace transform’s utility in solving complex summation problems, then develop a generalized frame- work of integral transforms that provides a systematic approach to evalu- ating nontrivial indefinite sums. Using the continuous binomial transform as a central example, we derive key identities that demonstrate the frame- work’s effectiveness. We also introduce a novel technique in addressing varying step sizes as well as nonlinear variable transformations in discrete indefinite summations. Our results show that integral transforms—with their flexible choice of kernels—serve as powerful tools for analyzing indefinite summations.