Binomial convolutions for rational power series

TL;DR

This paper introduces efficient algebraic methods for computing binomial convolutions and Hadamard products of rational power series, with applications to Fibonacci, tribonacci, and related sequences.

Contribution

It presents novel computational techniques using resultants, symmetric functions, and partial fractions for rational generating functions.

Findings

Efficient method using resultants for binomial convolution

Application to Fibonacci and tribonacci sequences

Alternative methods with symmetric functions and partial fractions

Abstract

The binomial convolution of two sequences $\{a_n\}$ and $\{b_n\}$ is the sequence whose $n$th term is $\sum_{k=0}^{n} \binom{n}{k} a_k b_{n-k}$. If $\{a_n\}$ and $\{b_n\}$ have rational generating functions then so does their binomial convolution. We discuss an efficient method, using resultants, for computing this rational generating function and give several examples involving Fibonacci and tribonacci numbers and related sequences. We then describe a similar method for computing Hadamard products of rational generating functions. Finally we describe two additional methods for computing binomial convolutions and Hadamard products of rational power series, one using symmetric functions and one using partial fractions.