A non-symmetric divide-and-conquer recursive formula for the convolution of polynomials and power series

TL;DR

This paper introduces a refined non-symmetric divide-and-conquer recursive formula for polynomial and power series convolution, utilizing classical mathematical sequences and transforms to improve clarity and applicability.

Contribution

It presents a new recursive convolution formula that leverages Sierpiński's polynomials, Thue-Morse sequence, and binomial modulo 2 transform, with multiple variants and a clearer notation.

Findings

Developed a cleaner, more conventional recursive convolution formula

Provided multiple variants and transformations of the formula

Enhanced understanding of polynomial and power series convolution methods

Abstract

Some changes in a recent convolution formula are performed here in order to clean it up by using more conventional notations and by making use of more referrenced and documented components (namely Sierpi\'nski's polynomials, the Thue-Morse sequence, the binomial modulo~2 transform and its inverse). Several variants are published here, by reading afterwards summed coefficients in another order; the last formula is then turned back from a summation to a new divide-and-conquer recursive formula.