Flattening Karatsuba's recursion tree into a single summation

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.