We study spectral moments of the Bures-Hall random matrices ensemble. The main result establishes a recurrence relation for the $k$-th spectral moment valid for a real-valued $k$, in contrast to prevailing results in the literature of different ensembles of assuming an integer $k$. The key to establish the recurrence relation is the obtained Christoffel-Darboux formulas of correlation kernels of the ensemble that avoid tedious summations. As an application of our spectral moment results, we re-derive the formulas of average von Neumann entropy and quantum purity of Bures-Hall ensemble conjectured by Ayana Sarkar and Santosh Kumar. This work is dedicated to the memory of Santosh Kumar.

翻译：本文研究了Bures-Hall随机矩阵系综的谱矩。主要结果建立了适用于实值$k$的第$k$阶谱矩的递推关系，这与文献中针对不同系综通常假设$k$为整数的现有结果形成对比。建立该递推关系的关键在于获得了该系综相关核的Christoffel-Darboux公式，从而避免了繁琐的求和运算。作为谱矩结果的应用，我们重新推导了Ayana Sarkar和Santosh Kumar所推测的Bures-Hall系综平均冯·诺依曼熵和量子纯度的计算公式。本工作谨献给Santosh Kumar以表纪念。