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Binomial convolutions for rational power series

Ira M. Gessel, Ishan Kar

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.

Binomial convolutions for rational power series

Abstract

The binomial convolution of two sequences and is the sequence whose th term is . If and 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.
Paper Structure (16 sections, 15 theorems, 130 equations, 1 table)

This paper contains 16 sections, 15 theorems, 130 equations, 1 table.

Table of Contents

  1. Introduction
  2. Computing binomial convolutions
  3. On rational power series over a field
  4. Rationality of binomial products
  5. Resultants
  6. The resultant of two polynomials
  7. Computing the binomial product with resultants
  8. Examples
  9. Fibonacci numbers
  10. Second order recurrent sequences
  11. Tribonacci numbers
  12. Perrin numbers
  13. The Hadamard product of rational power series
  14. Symmetric functions
  15. Partial fractions
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that where each $a_n$ is in a field $F$, and $p(x)$ and $q(x)$ are polynomials with coefficients in an extension field $G$ of $F$. Then there exist polynomials $P(x)$ and $Q(x)$ with coefficients in $F$ such that

Theorems & Definitions (26)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • proof : Proof of Theorem \ref{['t-fields']}
  • Lemma 4
  • proof
  • Corollary 5
  • proof
  • ...and 16 more