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)

A relationship between partial derivatives of the associated Legendre function of the first kind with respect to its degree, [ ∂ P ν m ( z ) / ∂ ν ] ν = n , and to its order, [ ∂ P n μ ( z ) / ∂ μ ] μ = m , is established for m , n ∈ N 0 . This relationship is used to deduce four new closed-form representations of [ ∂ P ν m ( z ) / ∂ ν ] ν = n from those found recently for [ ∂ P n μ ( z ) / ∂ μ ] μ = m by the author [R. Szmytkowski, On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order), J. Math. Chem. 46 (2009) 231]. Several new expressions for the associated Legendre function of the second kind of integer degree and order, Q n m ( z ) , suitable for numerical purposes, are also derived.