Recently, special functions of fractional order calculus have had many applications in various areas of mathematical analysis, physics, probability theory, optimization theory, graph theory, control systems, earth sciences, and engineering. Very recently, Zayed et al. (Mathematics 8:136, 2020) introduced the shifted Legendre-type matrix polynomials of arbitrary fractional orders and their various applications utilizing Rodrigues matrix formulas. In this line of research, we use the fractional order of Rodrigues formula to provide further investigation on such Legendre polynomials from a different point of view. Some properties, such as hypergeometric representations, continuation properties, recurrence relations, and differential equations, are derived. Moreover, Laplace’s first integral form and orthogonality are obtained.

最近，分数阶微积分的特殊功能已在数学分析，物理学，概率论，优化理论，图论，控制系统，地球科学和工程学的各个领域中得到了许多应用。最近，Zayed等人。（数学8：136，2020）介绍了任意分数阶的移位的Legendre型矩阵多项式及其利用Rodrigues矩阵公式的各种应用。在这方面的研究中，我们使用Rodrigues公式的分数阶来从不同的角度进一步研究此类Legendre多项式。导出了某些属性，例如超几何表示，连续性，递归关系和微分方程。此外，获得了拉普拉斯的第一积分形式和正交性。