A novel recurrence formula for moments with respect to M\"{u}ntz-Legendre polynomials is proposed and applied to construct a numerical method for solving generalized Gauss quadratures with power function weight for M\"{u}ntz systems. These quadrature rules exhibit several properties similar to the classical Gaussian quadratures for polynomial systems, including positive weights, rapid convergence, and others. They are applicable to a wide range of functions, including smooth functions and functions with endpoint singularities, commonly found in integral equations with singular kernels, complex analysis, potential theory, and other areas.

翻译：针对Müntz-Legendre多项式矩提出了一种新型递推公式，并将其应用于构造求解含幂函数权重的Müntz系统广义Gauss求积的数值方法。这些求积规则展现出与经典多项式系统Gauss求积相似的若干性质，包括正权重、快速收敛等特征。它们适用于广泛类型的函数，包括光滑函数及具有端点奇异性的函数，这类函数常见于含奇异核的积分方程、复分析、位势理论及其他领域。