Derivations and identities for the Chebyshev polynomials
AbstractUDC 519.114; 512.622
We introduce the notion of Chebyshev derivations of the first and second kinds based on the polynomial algebra and
corresponding specific differential operators, derive the elements of their kernels, and prove that any element of the kernel
of the derivations defines a polynomial identity for Chebyshev polynomials of both kinds. We obtain several polynomial
identities involving the Chebyshev polynomials of both kinds, a partial case of the Jacobi polynomials, and the generalized
L. Fox, I. B. Parker, Chebyshev polynomials in numerical analisys, Oxford Math. Handbooks, 3 (1968).
J. C. Mason, D. C. Handcomb, Chebyshev polynomials, Chapman and Hall/CRC, 3 (2002). DOI: https://doi.org/10.1201/9781420036114
L. Bedratyuk, Semi-invariants of binary forms and identities for Bernoulli, Euler and Hermite polynomials, Acta Arith., 151, 361 – 376 (2012), https://doi.org/10.4064/aa151-4-2 DOI: https://doi.org/10.4064/aa151-4-2
H. Prodinger, Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions, Open Math., 15, 1156 – 1160 (2017), https://doi.org/10.1515/math-2017-0096 DOI: https://doi.org/10.1515/math-2017-0096
E. D. Rainville, Special functions, Macmillan Co., New York (1960).
L. Ronald, L. Graham, D. E. Knuth, O. Patashnik, Concrete mathematics, Addison-Wesley, Reading, Massachusetts (1994).
G. Freudenburg, Algebraic theory of locally nilpotent derivations, Subseries: Invariant Theory and Algebraic Transformation Groups, Encyclopaedia Math. Sci., 136, № 7 (2017), https://doi.org/10.1007/978-3-662-55350-3 DOI: https://doi.org/10.1007/978-3-662-55350-3
A. A. Nowicki, Polynomial derivations and their rings of constants, Nicolaus Copernicus Univ. Press, Torun (1994).
J. J. Quaintance, H. Gould, Combinatorial identities for Stirling numbers: the unpublished notes of H W gould, Singapore, World Sci. Publ. (2016). DOI: https://doi.org/10.1142/9821
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