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

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 $D^{k}f(x),D^{k+1}f(x),\dots ,D^{2n+k+1}f(x)$, $k\in \mathbb{N}$ 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.