For the associated Legendre and Ferrers functions of the first and second kind, we obtain new multi-derivative and multi-integral representation formulas. The multi-integral representation formulas that we derive for these functions generalize some classical multi-integration formulas. As a result of the determination of these formulae, we compute some interesting special values and integral representations for certain particular combinations of the degree and order, including the case where there is symmetry and antisymmetry for the degree and order parameters. As a consequence of our analysis, we obtain some new results for the associated Legendre function of the second kind, including parameter values for which this function is identically zero.

中文翻译：

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关联 Legendre 和 Ferrers 函数的多积分表示

对于关联的第一类和第二类勒让德函数和费勒斯函数，我们获得了新的多导数和多积分表示公式。我们为这些函数推导出的多重积分表示公式推广了一些经典的多重积分公式。作为确定这些公式的结果，我们计算了一些有趣的特殊值和积分表示，用于次数和阶次的某些特定组合，包括次数和阶次参数存在对称和反对称的情况。作为我们分析的结果，我们获得了相关的第二类勒让德函数的一些新结果，包括该函数完全为零的参数值。