Preprint Article Version 1 This version is not peer-reviewed

Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers

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)

How to cite: Qi, F.; Guo, B. Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers. Preprints 2017, 2017030209 (doi: 10.20944/preprints201703.0209.v1). Qi, F.; Guo, B. Identities of the Chebyshev Polynomials, the Inverse of a Triangular Matrix, and Identities of the Catalan Numbers. Preprints 2017, 2017030209 (doi: 10.20944/preprints201703.0209.v1).

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.

Subject Areas

identity; inverse matrix; explicit formula; generating function; Chebyshev polynomials of the second kind; Catalan number; triangular matrix; classical hypergeometric function; integral representation

Readers' Comments and Ratings (2)

Comment 1
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
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Comment 2
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)
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