On the linear (in)dependence of sequences of derivatives of the functions $x^n\sin x$ and $x^n\cos x$

Jozef Fecenko; Enno Diekema

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arXiv · 2305.11184

On the linear (in)dependence of sequences of derivatives of the functions $x^n\sin x$ and $x^n\cos x$

Jozef Fecenko·Enno Diekema

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Abstract

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.

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BibTeX RIS

Jozef Fecenko, Enno Diekema. 2023-05-17. On the linear (in)dependence of sequences of derivatives of the functions $x^n\sin x$ and $x^n\cos x$. https://arxiv.org/abs/2305.11184

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