Abstract

In this paper, the authors investigate the case of discrete multiple orthogonal polynomials with two weights on the step line, which satisfy Pearson equations. The discrete multiple orthogonal polynomials in question are expressed in terms of \(\tau \) -functions, which are double Wronskians of generalized hypergeometric series. The shifts in the spectral parameter for type II and type I multiple orthogonal polynomials are described using banded matrices. It is demonstrated that these polynomials offer solutions to multicomponent integrable extensions of the nonlinear Toda equations. Additionally, the paper characterizes extensions of the Nijhoff–Capel totally discrete Toda equations. The hypergeometric \(\tau \) -functions are shown to provide solutions to these integrable nonlinear equations. Furthermore, the authors explore Laguerre–Freud equations, nonlinear equations for the recursion coefficients, with a particular focus on the multiple Charlier, generalized multiple Charlier, multiple Meixner II, and generalized multiple Meixner II cases.

中文翻译：

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超几何离散多重正交多项式的 Toda 和 Laguerre-Freud 方程和 tau 函数

摘要

在本文中，作者研究了在步长线上具有两个权重的离散多重正交多项式的情况，该多项式满足皮尔逊方程。所讨论的离散多重正交多项式用\(\tau \)函数表示，该函数是广义超几何级数的双朗斯基函数。使用带状矩阵描述 II 型和 I 型多重正交多项式的谱参数的变化。事实证明，这些多项式为非线性 Toda 方程的多分量可积扩展提供了解。此外，本文还描述了 Nijhoff-Capel 完全离散 Toda 方程的扩展。超几何\(\tau \)函数为这些可积非线性方程提供了解。此外，作者还探讨了拉盖尔-弗洛伊德方程、递归系数的非线性方程，特别关注多重 Charlier、广义多重 Charlier、多重 Meixner II 和广义多重 Meixner II 情况。