Bedratyuk, L. P.; Lunio, N. B. Derivations and identities for Chebyshev polynomials. (English. Ukrainian original) Zbl 1486.11038 Ukr. Math. J. 73, No. 8, 1175-1188 (2022); translation from Ukr. Mat. Zh. 73, No. 8, 1011-1022 (2021). Authors’ abstract: We introduce the notion of Chebyshev derivations of the first and second kinds based on the polynomial algebra and the corresponding specific differential operators, find the elements of their kernels, and prove that any element of the kernel of each derivation specifies a polynomial identity for Chebyshev polynomials of both kinds. We deduce several polynomial identities for the Chebyshev polynomials of both kinds, for a partial case of Jacobi polynomials, and for the generalized hypergeometric function. Reviewer: Klaus Schiefermayr (Wels) MSC: 11B83 Special sequences and polynomials 05A19 Combinatorial identities, bijective combinatorics 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) Keywords:Chebyshev polynomial PDFBibTeX XMLCite \textit{L. P. Bedratyuk} and \textit{N. B. Lunio}, Ukr. Math. J. 73, No. 8, 1175--1188 (2022; Zbl 1486.11038); translation from Ukr. Mat. Zh. 73, No. 8, 1011--1022 (2021) Full Text: DOI References: [1] L. Fox and I. B. Parker, Chebyshev Polynomials in Numerical Analysis, Oxford Math. Handbooks, 3, Oxford Univ. Press (1968). · Zbl 0153.17502 [2] J. C. Mason and D. C. Handcomb, Chebyshev Polynomials, Chapman & Hall/CRC, 3 (2002). [3] Bedratyuk, L., Semi-invariants of binary forms and identities for Bernoulli, Euler and Hermite polynomials, Acta Arith., 151, 361-376 (2012) · Zbl 1257.11020 [4] Prodinger, H., Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions, Open Math., 15, 1156-1160 (2017) · Zbl 1427.11017 [5] Rainville, ED, Special Functions (1960), New York: Macmillan Co., New York · Zbl 0092.06503 [6] Ronald, L.; Graham, L.; Knuth, DE; Patashnik, O., Concrete Mathematics (1994), Reading: Addison-Wesley, Reading · Zbl 0836.00001 [7] G. Freudenburg, Algebraic Theory of Locally Nilpotent Derivations, Subseries: Invariant Theory and Algebraic Transformation Groups, Encyclopedia Math. Sci., 136, No. 7, Springer, Berlin (2017). · Zbl 1391.13001 [8] Nowicki, AA, Polynomial Derivations and Their Rings of Constants (1994), Torun: Nicolaus Copernicus Univ. Press, Torun · Zbl 1236.13023 [9] http://www.dymoresolutions.com/UsersManual/Appendices/ChebyshevPolynomials.pdf, 4. [10] J. J. Quaintance and H. Gould, Combinatorial Identities for Stirling Numbers: the Unpublished Notes of H.W. Gould,World Scientific Publishing, Singapore (2016). · Zbl 1343.11002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.