A definition of d-dimensional n-Meixner random vectors is given first. This definition involves the commutators of their semi-quantum operators. After that we focus on the 1-Meixner random vectors and derive a system of d partial differential equations satisfied by their Laplace transform. We provide a set of necessary conditions for this system to be integrable. We use these conditions to give a complete characterization of all non-degenerate three-dimensional 1-Meixner random vectors. It must be mentioned that the three-dimensional case produces the first example in which the components of a 1-Meixner random vector cannot be reduced, via an injective linear transformation, to three independent classic Meixner random variables.

中文翻译：

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1-Meixner 随机向量

首先给出d维n-Meixner随机向量的定义。该定义涉及其半量子算子的交换子。之后，我们将重点放在 1-Meixner 随机向量上，并推导出满足其拉普拉斯变换的 d 个偏微分方程组。我们为该系统可集成提供了一组必要条件。我们使用这些条件给出所有非退化三维 1-Meixner 随机向量的完整表征。必须提到的是，三维情况产生了第一个示例，其中 1-Meixner 随机向量的分量不能通过内射线性变换减少为三个独立的经典 Meixner 随机变量。