Binomial convolutions for rational power series
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913232361357312 |
|---|---|
| author | Gessel, Ira M. Kar, Ishan |
| author_facet | Gessel, Ira M. Kar, Ishan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_10426 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Binomial convolutions for rational power series Gessel, Ira M. Kar, Ishan Combinatorics 05A19 (Primary) 05A15 (Secondary) 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. |
| title | Binomial convolutions for rational power series |
| topic | Combinatorics 05A19 (Primary) 05A15 (Secondary) |
| url | https://arxiv.org/abs/2304.10426 |