Hypergeometric-Type Sequences
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910414952988672 |
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| author | Tabuguia, Bertrand Teguia |
| author_facet | Tabuguia, Bertrand Teguia |
| contents | We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of sequences in this class are those defined by trigonometric functions with linear arguments in the index and $π$, such as Chebyshev polynomials, $\left(\sin^2\left(n\,π/4\right)\cdot\cos\left(n\,π/6\right)\right)_n$, and compositions like $\left(\sin\left(\cos(nπ/3)π\right)\right)_n$.
We describe an algorithm that computes a hypergeometric-type normal form of a given holonomic $n\text{th}$ term whenever it exists. Our implementation enables us to generate several identities for terms defined via trigonometric functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00256 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hypergeometric-Type Sequences Tabuguia, Bertrand Teguia Symbolic Computation Discrete Mathematics 33F10, 39A06 (Primary), 68R01, 05A19 (Secondary) I.1.1; I.1.2; G.2.1; G.4 We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of sequences in this class are those defined by trigonometric functions with linear arguments in the index and $π$, such as Chebyshev polynomials, $\left(\sin^2\left(n\,π/4\right)\cdot\cos\left(n\,π/6\right)\right)_n$, and compositions like $\left(\sin\left(\cos(nπ/3)π\right)\right)_n$. We describe an algorithm that computes a hypergeometric-type normal form of a given holonomic $n\text{th}$ term whenever it exists. Our implementation enables us to generate several identities for terms defined via trigonometric functions. |
| title | Hypergeometric-Type Sequences |
| topic | Symbolic Computation Discrete Mathematics 33F10, 39A06 (Primary), 68R01, 05A19 (Secondary) I.1.1; I.1.2; G.2.1; G.4 |
| url | https://arxiv.org/abs/2401.00256 |