对于Pearson（Ord）族的绝对连续（整数值）r. v. $X$，我们证明了在自然矩条件下，阶为$k$的Stein型协方差恒等式成立（cf. [Goldstein and Reinert，J. Theoret. Probab. 18（2005）237- 260]）。这个恒等式与由Rodrigues型公式得到的相应的正交多项式序列密切相关，并为任意函数的傅里叶系数提供了方便的表达式。协方差恒等式的应用为r. v. $X$的函数的相应的方差下界产生了一些新的表达式，这些表达式似乎仅在特定情况下才已知（对于正态，参见[Houdré and Kagan，J. Theoret. Probab. 8（1995）23- 30];另见[Houdré和Pérez-Abreu，Ann. Probab. 23（1995）400- 419]中关于Wiener和Poisson过程的相应结果）。并给出了一些应用实例。
英文摘要：
For an absolutely continuous (integer-valued) r.v. $X$ of the Pearson (Ord) family, we show that, under natural moment conditions, a Stein-type covariance identity of order $k$ holds (cf. [Goldstein and Reinert, J. Theoret. Probab. 18 (2005) 237--260]). This identity is closely related to the corresponding sequence of orthogonal polynomials, obtained by a Rodrigues-type formula, and provides convenient expressions for the Fourier coefficients of an arbitrary function. Application of the covariance identity yields some novel expressions for the corresponding lower variance bounds for a function of the r.v. $X$, expressions that seem to be known only in particular cases (for the Normal, see [Houdré and Kagan, J. Theoret. Probab. 8 (1995) 23--30]; see also [Houdré and Pérez-Abreu, Ann. Probab. 23 (1995) 400--419] for corresponding results related to the Wiener and Poisson processes). Some applications are also given.
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