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A relationship between partial derivatives of the associated Legendrefunction of the first kind with respect to its degree, $[\partialP_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$, and to its order, $[\partialP_{n}^{\mu}(z)/\partial\mu]_{\mu=m}$, is established for $m,n\in\mathbb{N}$.This relationship is used to deduce four new closed-form representations of$[\partial P_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$ from those found recently for$[\partial P_{n}^{\mu}(z)/\partial\mu]_{\mu=m}$ by the present author [R.Szmytkowski, J. Math. Chem. 46 (2009) 231]. Several new expressions for theassociated Legendre function of the second kind of integer degree and order,$Q_{n}^{m}(z)$, suitable for numerical purposes, are also derived.

On parameter derivatives of the associated Legendre function of the  first kind (with applications to the construction of the associated Legendre  function of the second kind of integer degree and order)