Power Series Composition and Change of Basis

TL;DR

This paper introduces efficient algorithms for power series composition and basis change, enabling fast conversions between various polynomial bases such as Euler, Bernoulli, Fibonacci, and orthogonal polynomials.

Contribution

It identifies a large class of power series for which composition is efficient and develops algorithms for basis conversion among many classical polynomial bases.

Findings

Fast algorithms for power series composition

Efficient basis change between multiple polynomial families

Applicable to Euler, Bernoulli, Fibonacci, and orthogonal polynomials

Abstract

Efficient algorithms are known for many operations on truncated power series (multiplication, powering, exponential, ...). Composition is a more complex task. We isolate a large class of power series for which composition can be performed efficiently. We deduce fast algorithms for converting polynomials between various bases, including Euler, Bernoulli, Fibonacci, and the orthogonal Laguerre, Hermite, Jacobi, Krawtchouk, Meixner and Meixner-Pollaczek.