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<p>This paper is dedicated to the 100th Birthday of V. S. Korolyuk, Academician of the National Academy of Sciences of Ukraine. 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). In addition, we analyze many other properties of these polynomials. In the inverse statement, we study the problem of existence of a bilinear functional for which two sequences of polynomials are given and form a biorthogonal system.</p>

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Biorthogonal Polynomials and Their Properties. Direct and Inverse Statements

  • 期刊论文

  • 发表日期：

    2025年12月6日

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  • Volodymyr Makarov

摘要

This paper is dedicated to the 100th Birthday of V. S. Korolyuk, Academician of the National Academy of Sciences of Ukraine. 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). In addition, we analyze many other properties of these polynomials. In the inverse statement, we study the problem of existence of a bilinear functional for which two sequences of polynomials are given and form a biorthogonal system.