Biorthogonal polynomials and their properties. Direct and inverse statements

Authors

  • V. Makarov Institute of Mathematics of the National Academy of Sciences of Ukraine, Kyiv

DOI:

https://doi.org/10.3842/umzh.v77i10.9381

Keywords:

completely positive operator, biorthogonal system of polynomials, Cauchy-type kernel, recurrence relation, differential equation, minimal order, integer sequences, generating function, common term of a sequence.

Abstract

UDC 510

This paper is dedicated to the centenary of Academician of the National Academy of Sciences of Ukraine V. S. Korolyuk. We consider a class of biorthogonal polynomials generated by a completely positive bilinear functional with Cauchy-type kernel (direct formulation). For these polynomials, we obtain four-term recurrence relations and differential equations of the minimal order (third and fourth). Moreover, we analyze many other properties of these polynomials. In the inverse statememt, we study the problem of existence of a bilinear functional for which two given sequences of polynomials form a biorthogonal system.

References

1. M. Bertola, M. Gekhman, J. Szmigielski, Caushy biorthogonal polynomials, J. Approx. Theory, 162, № 4, 832--867 (2010); https://doi.org/10.1016/j.jat.2009.09.008. DOI: https://doi.org/10.1016/j.jat.2009.09.008

2. M. Bertola, Biorthogonal polynomials for two-matrix models with semiclassical potentials, J. Approx. Theory, 144, № 2, 162--212 (2007); https://doi.org/10.1016/j.jat.2006.05.006. DOI: https://doi.org/10.1016/j.jat.2006.05.006

3. The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org.

4. Г. Бейтмен, А. Эрдейи, Высшие трансцендентные функции, Т. 2, Наука, Москва (1974).

5. В. Л. Макаров, Н. В. Майко, В. Л. Рябічев, Знаходження рекурентного співвідношення для системи поліномів у задачі з дробовою похідною, Кібернетика та системний аналіз, 61, № 1, 66–80 (2025); DOI 10.34229/kca2522-9664.25.1.7.

6. W. Hahn, Über die Jacobischen Polynome und zwei verwandte Polynomklassen, Math. Z., 39, 634–638 (1935). DOI: https://doi.org/10.1007/BF01201380

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Published

30.09.2025

Issue

Vol. 77 No. 10 (2025)

Section

Research articles

License

Copyright (c) 2025 Володимир Макаров

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Makarov, V. “Biorthogonal Polynomials and Their Properties. Direct and Inverse Statements”. Ukrains’kyi Matematychnyi Zhurnal, vol. 77, no. 10, Sept. 2025, pp. 629–639, https://doi.org/10.3842/umzh.v77i10.9381.