2019/10/24 by Leonid Bedratyuk, Bedratyuk, Leonid, Nataliia Luno +1
Mathematics · Physics and Astronomy · #11B83 #33C45 #Advanced Mathematical Identities #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics
paper · pdf · doi:10.48550/arxiv.1910.10990
openalex publication_date 2019/10/24 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
In this paper we follow the general approach, proposed earlier by the first\nauthor, which is derived from the invariant theory field and provides a way of\nobtaining of the polynomial identities for any arbitrary polynomial family. We\nintroduce the notion of Chebyshev derivations of the first and second kinds,\nwhich is based on the polynomial algebra, and corresponding specific\ndifferential operators. We derive the elements of their kernels and prove that\nany element of the kernel of the derivations defines a polynomial identity\nsatisfied by the Chebyshev polynomials of the first and second kinds. Combining\nelementary methods and combinatorial techniques, we obtain several new\npolynomial identities involving the Chebyshev polynomials of the both kinds and\na special case of the Jacobi polynomials. Using the properties of the\ngeneralised hypergeometric function, we specify the Chebyshev polynomials of\nthe first and second kinds via the generalised hypergeometric function and, as\na consequence, derive the corresponding identities involving the generalised\nhypergeometric function and the Chebyshev polynomials of the first and second\nkinds.\n