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.
Received:
31 August 2017
Commenter:
Feng Qi
Commenter's Conflict of Interests:
I am the first and corresponding author of this preprint.
Comment:
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 and Chinese presentation 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)
Received:
5 November 2019
Commenter:
Feng Qi
Commenter's Conflict of Interests:
I am the first and corresponding author
Comment:
This preprint has been formally published as
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 13 (2019), no. 2, 518--541; available online at https://doi.org/10.2298/AADM190118018Q
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the first and corresponding author of this preprint.
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)
Commenter: Feng Qi
Commenter's Conflict of Interests: I am the first and corresponding author
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 13 (2019), no. 2, 518--541; available online at https://doi.org/10.2298/AADM190118018Q