An extension of the Abel–Liouville identity
Abstract
In this note, we present an extension of the celebrated Abel–Liouville identity in terms of noncommutative complete Bell polynomials for generalized Wronskians. We also characterize the range equivalence of n-dimensional vector-valued functions in the subclass of n-times differentiable functions with a nonvanishing Wronskian.
Keywords
Abel–Liouville identity; generalized Wronskians; complete Bell polynomials
References
E.T. Bell, Partition polynomials, Ann. of Math. (2) 29 (1927/28), no. 1–4, 38–46.
E.T. Bell, Exponential polynomials, Ann. of Math. (2) 35 (1934), no. 2, 258–277.
R. Schimming and S.Z. Rida, Noncommutative Bell polynomials, Internat. J. Algebra Comput. 6 (1996), no. 5, 635–644.
G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, 140, American Mathematical Society, Providence, RI, 2012.
Institute of Mathematics, University of Debrecen, Hungary Hungary
https://orcid.org/0000-0003-2382-6035
Department of Mathematics, Faculty of Education, Ain Shams University, Egypt Egypt
amr.zakaria@edu.asu.edu.eg
This work is licensed under a Creative Commons Attribution 4.0 International License.
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
