The article presents a method to calculate moments of order statistics more efficiently by using recurrence relations and identities. By applying this method, researchers can reduce the amount of calculations needed and improve accuracy. Previous studies have shown similar relations for specific distributions like exponential, uniform, Pareto, and logistic. The new approach extends these findings to include power function, Pareto, Weibull, Burr XII, and logistic distributions, providing a way to calculate moments for distributions with at least two parameters and non-integral orders. This method also reveals structural properties and relationships between different probability distributions.