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<p>In Vinet and Zhedanov (J Phys A Math Theor 45:265304, 2012), while looking for spin chains that admit perfect state transfer, Vinet and Zhedanov found an apparently new sequence of orthogonal polynomials, that they called para-Krawtchouk polynomials, defined on a bilinear lattice. In this note we present necessary and sufficient conditions for the regularity of solutions of the corresponding functional equation. Moreover, the functional Rodrigues formula and a closed formula for the recurrence coefficients are presented. As a consequence, we characterize all solutions of the functional equation, including as very particular cases the Meixner, Charlier, Krawtchouk, Hahn, and para-Krawtchouk polynomials.</p>

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On classical orthogonal polynomials on bi-lattices

  • 期刊论文

  • 发表日期：

    2025年2月5日

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  • K. Castillo,
  • G. Filipuk,
  • D. Mbouna

摘要

In Vinet and Zhedanov (J Phys A Math Theor 45:265304, 2012), while looking for spin chains that admit perfect state transfer, Vinet and Zhedanov found an apparently new sequence of orthogonal polynomials, that they called para-Krawtchouk polynomials, defined on a bilinear lattice. In this note we present necessary and sufficient conditions for the regularity of solutions of the corresponding functional equation. Moreover, the functional Rodrigues formula and a closed formula for the recurrence coefficients are presented. As a consequence, we characterize all solutions of the functional equation, including as very particular cases the Meixner, Charlier, Krawtchouk, Hahn, and para-Krawtchouk polynomials.