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arXiv · 2601.12579

A semigroup approach to iterated binomial transforms

Johann Verwee
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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.

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Johann Verwee. 2026-01-22. A semigroup approach to iterated binomial transforms. https://arxiv.org/abs/2601.12579

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