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<p>We study a new class of special functions called Laguerre–Bessel polynomials (LBPs), which play the same role as Laguerre polynomials in the case of application of the Cayley transform to the evaluation of the operator exponent. The generating function for the LBPs is a Bessel operator function of the firstkind of order zero obtained by replacing the operator argument by its Cayley transform. It is proved that the LBPs coincide with a new class of 2-orthogonal polynomials with an accuracy to within a linear transformation. It is also shown that the LBPs are classical in the Hahn–Maroni sense because their normalized derivatives also form a class of 2-orthogonal polynomials with an accuracy to within a linear transformation. We also determine the following characteristics traditional for the polynomial classes: explicit representation, recurrence relations, and the differential equation of the minimal third order.</p>

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Riemann Operator Function. Cayley’s Transform

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • Volodymyr Makarov

摘要

We study a new class of special functions called Laguerre–Bessel polynomials (LBPs), which play the same role as Laguerre polynomials in the case of application of the Cayley transform to the evaluation of the operator exponent. The generating function for the LBPs is a Bessel operator function of the firstkind of order zero obtained by replacing the operator argument by its Cayley transform. It is proved that the LBPs coincide with a new class of 2-orthogonal polynomials with an accuracy to within a linear transformation. It is also shown that the LBPs are classical in the Hahn–Maroni sense because their normalized derivatives also form a class of 2-orthogonal polynomials with an accuracy to within a linear transformation. We also determine the following characteristics traditional for the polynomial classes: explicit representation, recurrence relations, and the differential equation of the minimal third order.