 [2305.11184] 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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Mathematics > General Mathematics


arXiv:2305.11184 (math)
    




  [Submitted on 17 May 2023 (v1), last revised 24 Jan 2024 (this version, v2)]
Title:On the linear (in)dependence of sequences of derivatives of the functions xnsinx and xncosx
Authors:Jozef Fecenko, Enno Diekema View a PDF of the paper titled On the linear (in)dependence of sequences of derivatives of the functions $x^n\sin x$ and $x^n\cos x$, by Jozef Fecenko and 1 other authors
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Abstract:The main goal of the paper is to prove that the sequence of functions f(x),Df(x),…,D2n+1f(x), where f(x) is xnsinx or xncosx are linearly independent. Or more generally: that the sequence of functions Dkf(x),Dk+1f(x),…,D2n+k+1f(x), k∈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.
    


 
Comments:
28 pages


Subjects:

General Mathematics (math.GM)
 
MSC classes:
34A30, 15A03, 05A19


ACM classes:
F.2.2


Cite as:
arXiv:2305.11184 [math.GM]


 
(or 
arXiv:2305.11184v2 [math.GM] for this version)
          
 
 

https://doi.org/10.48550/arXiv.2305.11184



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                arXiv-issued DOI via DataCite
              







Submission history From: Jozef Fecenko [view email]       [v1]
        Wed, 17 May 2023 19:17:13 UTC (16 KB)
[v2]
        Wed, 24 Jan 2024 15:01:48 UTC (18 KB)



 

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