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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

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.

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

Abstract

The main goal of the paper is to prove that the sequence of functions , where is or are linearly independent. Or more generally: that the sequence of functions , 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.
Paper Structure (5 sections, 15 theorems, 129 equations)

This paper contains 5 sections, 15 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. A method using the Wronskian
  3. A method using the definition of linear (in)dependence of functions and combinatorial identities
  4. Apendix
  5. Conclusion

Key Result

Lemma 1

Let $\hbox{\boldmath$A$}=[a_{ij}]_{n\times n}$ and $k=1,2,\dots,n-1, n=2,3,\dots$ then $\hbox{\boldmath$R$}_k^T$ is the transposed matrix to the matrix $\hbox{\boldmath$R$}_k$.

Theorems & Definitions (32)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 22 more