A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable
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A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable Ol’ga V. Veselovska, Veronika V. Dostoina (Presented by V. O. Derkach) Abstract. For the derivatives of Chebyshev second-kind polynomials of a complex vafiable, a system of functions biorthogonal with them on closed curves of the complex plane is constructed. Properties of these functions and the conditions of expansion of analytic functions in series in polynomials under consideration are established. The examples of such expansions are given. In addition, we obtain some combinatorial identities of independent interest. Keywords. Chebyshev second-kind polynomials, analytic function, biorthogonal systems of functions, associated function.
Introduction The properties of orthogonal systems of polynomials of a real variable and the expansions of functions in series in them were sufficiently fully studied in the scientific literature, in particular, in [1, 2]. A significantly less number of works deal with properties of those systems in complex domains. In work [2], some properties of Chebyshev polynomials of a complex variable are considered, and the expansions of analytic functions in Chebyshev first-kind polynomials in complex domains are presented. The properties of polynomials related to Chebyshev polynomials and the expansions of analytic functions in series in them in complex domains were studied in [8, 9], respectively. A generalization of the method of expansion of functions in power series is their expansion in a system of polynomials biorthogonal with some other system of functions called associated. Under definite conditions for any linearly independent and complete system of functions, it is possible to construct the corresponding system of associated functions and to obtain series in them. In the present work, we will construct a system of associated functions biorthogonal with the derivatives of Chebyshev second-kind polynomials on closed curves of the complex plane and find the conditions under which the analytic functions can be expanded in series in those polynomials. We will give some examples of such expansions. In addition, we obtain combinatorial identities which can be used in other studies.
1.
Derivatives of Chebyshev second-kind polynomials of a complex variable
Let Cnm be binomial coefficients, and let Un (z) be Chebyshev second-kind polynomials of a complex variable. The polynomials Un (z) satisfy the relation [3]: [ n2 ] ∑ k (−1)k 2n−2k Cn−k Un (z) = z n−2k , k=0
Translated from Ukrains’ki˘ı Matematychny˘ı Visnyk, Vol. 17, No. 2, pp. 256–277 April–June, 2020. Original article submitted February 11, 2020
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(1.1)
where n = 0, 1, . . .. Let us differentiate relation (1.1) s times: [
Un(s) (z) =
n−s 2 ∑
] k (−1)k 2n−2k(n − 2k)(n − 2k − 1). . .(n − 2k − s + 1)Cn−k z n−2k−s ,
k=0
where s ≤ n. Since s (n − 2k)(n − 2k − 1) . . . (n − 2k − s + 1) = s!Cn−2k , (s)
we get the formula for the s-th derivative Un (z)
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