In this note, we derive the closed form formulae for moments of Student's t-distribution in the one dimensional case as well as in higher dimensions through a unified probability framework. Interestingly, the closed form expressions for the moments of Student's t-distribution can be written in terms of the familiar Gamma function, Kummer's confluent hypergeometric function, and the hypergeometric function.

翻译：本文通过统一的概率框架，推导出一维及高维情形下学生t分布矩的闭式表达式。值得注意的是，学生t分布矩的闭式表达式可以用常见的Gamma函数、库默尔合流超几何函数以及超几何函数表示。