A semigroup approach to iterated binomial transforms
Johann Verwee
TL;DR
This work introduces a one-parameter binomial-convolution operator $T_r$ that forms an additive semigroup via $T_r \circ T_s = T_{r+s}$ and has inverse $T_{-r}$, unifying iterated binomial transforms under a compact algebraic structure. Generating functions reveal closed forms, with $A_r(z)=\frac{1}{1-rz} A\left(\frac{z}{1-rz}\right)$ and $\widehat{A}_r(t)=e^{rt}\widehat{A}(t)$, and the Riordan-array representation $T_r = \big((1-rz)^{-1},\; z(1-rz)^{-1}\big)$. A central root-shift principle shows that applying $T_r$ translates the roots of constant-coefficient recurrences by $r$, turning $P(S)\mathbf{a}=0$ into $P(S-r)\mathbf{b}=0$, with explicit formulas for second-order and Binet-type representations. The paper provides a second-order template and corollaries for classic sequences (Fibonacci, Lucas, Pell, Jacobsthal, Mersenne), plus OEIS-friendly data and tables, and offers two intuitive interpretations: a combinatorial coloring view via exponential generating functions and a linear-algebraic view as a spectral shift $M\ mapsto M+rI$. Overall, the framework streamlines the analysis of iterated binomial transforms and points to natural extensions within the Riordan/Sheffer calculus.
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.