On the linear (in)dependence of sequences of derivatives of the functions $x^n\sin x$ and $x^n\cos x$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916104110080000 |
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| author | Fecenko, Jozef Diekema, Enno |
| author_facet | Fecenko, Jozef Diekema, Enno |
| contents | The main goal of the paper is to prove that the sequence of functions $f(x), Df(x), \dots, D^{2n+1}f(x)$, where $f(x)$ is $x^n\sin x$ or $x^n\cos x$ are linearly independent. Or more generally: that the sequence of functions $D^kf(x), D^{k+1}f(x), \dots, D^{2n+k+1}f(x)$, $k\in \mathbb{N}$ is linearly independent. The problem is solved by a suitable transformation of the matrix of determinant of the Wronskian. Another approach for a special sequence of derivatives of functions uses only the definition of linear independence of functions. This approach generates interesting, non-elementary combinatorial identities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_11184 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the linear (in)dependence of sequences of derivatives of the functions $x^n\sin x$ and $x^n\cos x$ Fecenko, Jozef Diekema, Enno General Mathematics 34A30, 15A03, 05A19 F.2.2 The main goal of the paper is to prove that the sequence of functions $f(x), Df(x), \dots, D^{2n+1}f(x)$, where $f(x)$ is $x^n\sin x$ or $x^n\cos x$ are linearly independent. Or more generally: that the sequence of functions $D^kf(x), D^{k+1}f(x), \dots, D^{2n+k+1}f(x)$, $k\in \mathbb{N}$ is linearly independent. The problem is solved by a suitable transformation of the matrix of determinant of the Wronskian. Another approach for a special sequence of derivatives of functions uses only the definition of linear independence of functions. This approach generates interesting, non-elementary combinatorial identities. |
| title | On the linear (in)dependence of sequences of derivatives of the functions $x^n\sin x$ and $x^n\cos x$ |
| topic | General Mathematics 34A30, 15A03, 05A19 F.2.2 |
| url | https://arxiv.org/abs/2305.11184 |