A semigroup approach to iterated binomial transforms

Abstract

We study a one-parameter family of binomial-convolution operators acting on sequences. These operators form an additive semigroup with an explicit inverse, and they subsume iterated classical binomial transforms as a special case. We describe the action in terms of ordinary and exponential generating functions, interpret the transform in the Riordan-array framework, and prove a general root-shift principle for constant-coefficient linear recurrences: applying the transform shifts the characteristic roots by a fixed amount. Several classical families (Fibonacci, Lucas, Pell, Jacobsthal, Mersenne) are treated uniformly as illustrative examples.