# Flattening Karatsuba's recursion tree into a single summation

## TL;DR

This paper introduces an interleaved splitting scheme for Karatsuba's recursion tree, simplifying access to nodes and enabling the flattening of the entire tree into a single convolution formula with fewer multiplications, applicable to infinite power series.

## Contribution

It presents a novel interleaved splitting method that flattens Karatsuba's recursion tree into a single summation, reducing elementary multiplications and extending applicability to infinite series.

## Key findings

- Flattened recursion tree into a single formula.
- Reduced number of elementary multiplications.
- Applicable to infinite power series.

## Abstract

The recursion tree resulting from Karatsuba's formula is built here by using an interleaved splitting scheme rather than the traditional left/right one. This allows an easier access to the nodes of the tree and $2n-1$ of them are initially flattened all at once into a single recursive formula. The whole tree is then flattened further into a convolution formula involving less elementary multiplications than the usual Cauchy product. Unlike the traditional splitting scheme, the interleaved approach may also be applied to infinite power series, and corresponding formulas are also given.