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<p>In <i>Lett. Math. Phys.</i> <b>114</b>, 54 (2024) and <b>115</b>, 70 (2025), the author introduces what is presented as a novel method for determining whether a sequence of orthogonal polynomials is “classical”, based solely on its initial recurrence coefficients. This note demonstrates that all the results contained in those works are already encompassed by two general theorems previously established in <i>J. Math. Anal. Appl.</i> <b>515</b> (2022), Article 126390. A symbolic algorithm, implemented in <span>Mathematica</span>, is also provided to enable automated verification of the classical character of orthogonal polynomial sequences on quadratic lattices. As an application, it is shown that the so-called para-Krawtchouk polynomials on bi-lattices, discussed in <i>Lett. Math. Phys.</i> <b>115</b>, 70 (2025), constitute a particular instance of a classical orthogonal family on a linear lattice. Consequently, their algebraic properties follow as a specific case of one of the main theorems established in <i>J. Math. Anal. Appl.</i> <b>515</b> (2022), Article 126390.</p>

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On recurrence coefficients of classical orthogonal polynomials

  • 期刊论文

  • 发表日期：

    2025年11月12日

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  • K. Castillo,
  • G. Gordillo-Núñez

摘要

In Lett. Math. Phys. 114, 54 (2024) and 115, 70 (2025), the author introduces what is presented as a novel method for determining whether a sequence of orthogonal polynomials is “classical”, based solely on its initial recurrence coefficients. This note demonstrates that all the results contained in those works are already encompassed by two general theorems previously established in J. Math. Anal. Appl. 515 (2022), Article 126390. A symbolic algorithm, implemented in Mathematica, is also provided to enable automated verification of the classical character of orthogonal polynomial sequences on quadratic lattices. As an application, it is shown that the so-called para-Krawtchouk polynomials on bi-lattices, discussed in Lett. Math. Phys. 115, 70 (2025), constitute a particular instance of a classical orthogonal family on a linear lattice. Consequently, their algebraic properties follow as a specific case of one of the main theorems established in J. Math. Anal. Appl. 515 (2022), Article 126390.