An Elementary Theory of Indefinite Summation Using Integral Transforms

We develop a generalized framework for a novel approach to indefinite summation through the use of integral transforms. Central to our development is the continuous binomial transform, through which we derive key identities that validate the consistency and effectiveness of the method. The framework further extends to accommodate variable step sizes and addresses the limitations of general nonlinear transformations of the summation index. Our results demonstrate that integral transforms are a powerful and flexible tool for the analysis and computation of discrete indefinite sums.