The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, quantum q-Krawtchouk, q-Krawtchouk, affine q-Krawtchouk, q-Charlier, Al-Salam–Carlitz, q-Hahn, little q-Jacobi, little q-Laguerre/Wall, q-Bessel, q-Racah, and dual q-Hahn polynomials.

中文翻译：

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经典离散正交多项式的斯蒂尔杰斯定理

本笔记的目的是根据 Nikiforov 和 Uvarov 引入的超几何型差分方程，为经典离散正交多项式的零点实参数的单调性建立新的易于处理的充分条件。这一结果允许对线性、二次、q-线性和q-二次网格上经典正交多项式的零点的单调性进行系统研究。特别是，我们以简单统一的方式分析了 Hahn、Charlier、Krawtchouk、Meixner、Racah、对偶 Hahn、q-Meixner、量子 q-Krawtchouk、q-Krawtchouk、仿射 q-Krawtchouk、q-的零点的单调性Charlier、Al-Salam-Carlitz、q-Hahn、little q-Jacobi、little q-Laguerre/Wall、q-Bessel、q-Racah 和对偶 q-Hahn 多项式。