We study the properties of the systems of polynomials of complex variable represented in the form of contour integrals with kernel functions analytic at infinity. We formulate the conditions for the existence of functions associated with these polynomials and sufficient conditions for the expansion of analytic functions in series in these polynomials. The accumulated results can be used to find the expansions of functions in series in the classical orthogonal polynomials in complex domains, the integral representations for some of these polynomials, the dependences of monomials zn of these polynomials, and other relations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. I. Markushevich, Selected Chapters of the Theory of Analytic Functions [in Russian], Nauka, Moscow (1976).
S. Pashkovskii, Computing Applications of Chebyshev Polynomials and Series [in Russian], Nauka, Moscow (1983).
G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis. Ester Band. Reihen. Integralrechnung. Funktionentheorie, Vol. 1 [Russian translation], Nauka, Moscow (1978).
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Elementary Functions [in Russian], Nauka, Moscow (1981).
G. Szegö, Orthogonal Polynomials [Russian translation], Fizmatgiz, Moscow (1962).
M. A. Sukhorol’s’kyi, “Expansion of functions in a system of polynomials biorthogonal on a closed contour with a system of functions regular at infinitely remote point,” Ukr. Mat. Zh., 62, No. 2, 238–254 (2010); English translation: Ukr. Math. J., 62, No. 2, 268–288 (2010).
M. A. Sukhorol’s’kyi, “Approximation of functions by Legendre polynomials in the complex plane,” Visn. Nats. Univ. “L’vivs’ka Politekhnika”, No. 643, 3–14 (2009).
M. A. Sukhorol’s’kyi and V. V. Dostoina, “Expansion of functions analytic in a disk in the complex plane in a system of derivatives of Legendre polynomials,” Visn. Nats. Univ. “L’vivs’ka Politekhnika”, No. 687, 105–121 (2010).
M. A. Sukhorol’s’kyi, V. V. Dostoina, and O. V. Veselovs’ka, “Polynomials related to Chebyshev polynomials,” Prikl. Probl. Mekh. Mat., 15, 35–41 (2017).
M. A. Sukhorol’s’kyi, V. V. Dostoina, and O. V. Veselovs’ka, “System of functions biorthogonal to polynomials related to the Chebyshev polynomials,” Visn. Kyiv. Nats. Univ., Mat. Mekh., No. 1, 53–59 (2018).
O. V. Veselovska and V. V. Dostoina, “A system of functions biorthogonal with the derivatives of Chebyshev second-kind polynomials of a complex variable,” J. Math. Sci., 249, No. 5, 785–803 (2020).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 11, pp. 1516–1531, November, 2021. Ukrainian DOI: https://doi.org/10.37863/umzh.v73i11.6699.
M. A. Sukhorolsky is deceased.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Sukhorolsky, M.A., Veselovska, O.V. & Dostoina, V.V. System of Polynomials of Complex Variable Related to the Classical Systems of Orthogonal Polynomials. Ukr Math J 73, 1752–1771 (2022). https://doi.org/10.1007/s11253-022-02028-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-022-02028-y