Lommel functions, Padé approximants and hypergeometric functions
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909232573448192 |
|---|---|
| author | Zullo, Federico |
| author_facet | Zullo, Federico |
| contents | We consider the Lommel functions $s_{μ,ν}(z)$ for different values of the parameters $(μ,ν)$. We show that if $(μ,ν)$ are half integers, then it is possible to describe these functions with an explicit combination of polynomials and trigonometric functions. The polynomials turn out to give Padé approximants for the trigonometric functions. Numerical properties of the zeros of the polynomials are discussed. Also, when $μ$ is an integer, $s_{μ,ν}(z)$ can be written as an integral involving an explicit combination of trigonometric functions. A closed formula for $_2F_1\left(\frac{1}{2}+ν,\frac{1}{2}-ν;μ+\frac{1}{2};\sin(\fracθ{2})^2\right)$ with $μ$ an integer is given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19359 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lommel functions, Padé approximants and hypergeometric functions Zullo, Federico Classical Analysis and ODEs Mathematical Physics We consider the Lommel functions $s_{μ,ν}(z)$ for different values of the parameters $(μ,ν)$. We show that if $(μ,ν)$ are half integers, then it is possible to describe these functions with an explicit combination of polynomials and trigonometric functions. The polynomials turn out to give Padé approximants for the trigonometric functions. Numerical properties of the zeros of the polynomials are discussed. Also, when $μ$ is an integer, $s_{μ,ν}(z)$ can be written as an integral involving an explicit combination of trigonometric functions. A closed formula for $_2F_1\left(\frac{1}{2}+ν,\frac{1}{2}-ν;μ+\frac{1}{2};\sin(\fracθ{2})^2\right)$ with $μ$ an integer is given. |
| title | Lommel functions, Padé approximants and hypergeometric functions |
| topic | Classical Analysis and ODEs Mathematical Physics |
| url | https://arxiv.org/abs/2406.19359 |