On classical orthogonal polynomials on bi-lattices
K. Castillo, G. Filipuk, D. Mbouna
TL;DR
This work extends the theory of classical orthogonal polynomials to bi-lattices by formulating a distributional Pearson-type equation with bi-lattice difference operators $\mathrm{D}_s$ and $\mathrm{S}_s$. It provides a sharp regularity criterion, a Rodrigues-type formula, and explicit three-term recurrence coefficients for the associated OPS, and classifies all regular solutions into two families $H_n^{\gamma}$ and $Q_n^{\gamma}$, showing that Meixner, Charlier, Krawtchouk, Hahn, and para-Krawtchouk appear as special cases. The main results, Theorems 1 and 2, establish necessary and sufficient regularity conditions and a complete degree-based classification (including para-Krawtchouk as a specialization of $Q_n^{\gamma}$), thus unifying classical lattice polynomials within a single bi-lattice framework. This advances the systematic identification of explicit OPS on bi-lattices with Rodrigues-type representations and clear recurrence structures, with potential applications in areas such as spin chains and quantum information where bi-lattice spectra arise.
Abstract
In [J. Phys. A: Math. Theor. 45 (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.