Supplement to the article by Douak and Maroni (2020) on a new class of 2-orthogonal polynomials
DOI:
https://doi.org/10.3842/umzh.v77i3.8898Keywords:
2-orthogonal polynomials, 2-orthogonal polynomials, generating function, recurrence relation, ``classical'' 2-orthogonal polynomials, differential equation, zeros of orthogonal polynomials.Abstract
UDC 510
We consider some issues related to the 2-orthogonal polynomials (2-OP). The answers to these issues can be regarded a supplement to the article [K. Douak, P. Maroni, On a new class of 2-orthogonal polynomials, I: The recurrence relations and some properties, Integral Transform and Special Functions (2020)]. The conditions imposed on the parameters of two original recurrence relations (the first of these conditions is for the 2-OP, while the second condition is for their normalized derivatives) and guaranteeing the ``"classical" nature of the 2-OP in Hahn's sense are clarified. It is constructively proved that these recurrence relations do not cover all possible "classical" 2-OPs. An example of ``"classical" 2-OP generated by the generating function constructed by using a Bessel function of the first kind of order zero is presented. These OPs are unique because their properties are similar to the classical OPs. In particular, this concerns the fact that their zeros are real and their location. The analysis of the available literature and our own numerous numerical-analytic experiments reveals the absence of other examples of the ``"classical" 2-OPs with similar properties.
References
1. K. Douak, P. Maroni, On a new class of 2-orthogonal polynomials, I: The recurrence relations and some properties, Integral Transform and Special Functions, 32, Issue 2, 134–153 (2021); https://doi.org/10.1080/ 10652469.2020.1811702. DOI: https://doi.org/10.1080/10652469.2020.1811702
2. K. Douak, P. Maroni, On a new class of 2-orthogonal polynomials, II. The integral representations, arXiv:2212.11949 [math.CA] (2022); https://doi.org/10.48550/arXiv.2212.11949.
3. W. Hahn, Über die Jacobischen Polynome und zwei verwandte Polynomklassen, Math. Z., 39, 634–638 (1935). DOI: https://doi.org/10.1007/BF01201380
4. В. Л. Макаров, Операторна функція Рімана. Перетворення Келі, Укр. мат. журн., 77, № 2, 139–143 (2025). DOI: https://doi.org/10.3842/umzh.v77i2.8859
5. Ana F. Loureiro, P. Maroni, S. Yakubovich, On a polynomial sequence associated with the Bessel operator, Proc. Amer. Math. Soc., 142, № 2, 467–482 (2014). DOI: https://doi.org/10.1090/S0002-9939-2013-11658-8
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Copyright (c) 2025 V. Makarov
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