Version 1
: Received: 27 March 2017 / Approved: 28 March 2017 / Online: 28 March 2017 (03:23:49 CEST)
Version 2
: Received: 7 August 2017 / Approved: 7 August 2017 / Online: 7 August 2017 (15:49:11 CEST)
Feng Qi, Qing Zou, and Bai-Ni Guo, The inverse of a triangular matrix and several identities of the Catalan numbers. Applicable Analysis and Discrete Mathematics, 2019, 13, 2, doi:10.2298/AADM190118018Q.
Feng Qi, Qing Zou, and Bai-Ni Guo, The inverse of a triangular matrix and several identities of the Catalan numbers. Applicable Analysis and Discrete Mathematics, 2019, 13, 2, doi:10.2298/AADM190118018Q.
Feng Qi, Qing Zou, and Bai-Ni Guo, The inverse of a triangular matrix and several identities of the Catalan numbers. Applicable Analysis and Discrete Mathematics, 2019, 13, 2, doi:10.2298/AADM190118018Q.
Feng Qi, Qing Zou, and Bai-Ni Guo, The inverse of a triangular matrix and several identities of the Catalan numbers. Applicable Analysis and Discrete Mathematics, 2019, 13, 2, doi:10.2298/AADM190118018Q.
Abstract
In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.
Keywords
identity; inverse matrix; explicit formula; generating function; Chebyshev polynomials of the second kind; Catalan number; triangular matrix; classical hypergeometric function; integral representation
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Commenter's Conflict of Interests:
I am the corresponding author of this preprint.
Comment:
This preprint should be cited as
Feng Qi, Qing Zou, and Bai-Ni Guo, Some identities and a matrix inverse related to the Chebyshev polynomials of the second kind and the Catalan numbers, Preprints 2017, 2017030209, 25 pages; Available online at http://dx.doi.org/10.20944/preprints201703.0209.v2.
Received:
13 October 2017
Commenter:
Feng Qi
Commenter's Conflict of Interests:
I am the first and corresponding author for this preprint
Comment:
A reorganized presentation in Chinese for this preprint has been posted as
Feng Qi, The inverse of a matrix and several identities related to the Catalan numbers and the Chebyshev polynomials of the second kind, ResearchGate Presentation (2017), available online at https://doi.org/10.13140/RG.2.2.22832.46081 (Chinese)
Commenter:
Commenter's Conflict of Interests: I am the corresponding author of this preprint.
Feng Qi, Qing Zou, and Bai-Ni Guo, Some identities and a matrix inverse related to the Chebyshev polynomials of the second kind and the Catalan numbers, Preprints 2017, 2017030209, 25 pages; Available online at http://dx.doi.org/10.20944/preprints201703.0209.v2.
For accurate information to cite this preprint, please download its PDF file from https://www.preprints.org/manuscript/201703.0209/v2/download
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the first and corresponding author for this preprint
Feng Qi, The inverse of a matrix and several identities related to the Catalan numbers and the Chebyshev polynomials of the second kind, ResearchGate Presentation (2017), available online at https://doi.org/10.13140/RG.2.2.22832.46081 (Chinese)