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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 $x^n\cos x$ arelinearly 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 linearlyindependent. The problem is solved by a suitable transformation of the matrixof determinant of the Wronskian. Another approach for a special sequence ofderivatives of functions uses only the definition of linear independence offunctions. This approach generates interesting, non-elementary combinatorialidentities.

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