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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). (English) Zbl 1232.33005
The author studies the associated Legendre functions of the first kind ${P}_{\nu }^{\mu }\left(z\right)$ in case of integer values of the degree $\nu$ and the order $\mu$. The main result is a relationship between the partial derivative ${\left[\partial {P}_{\nu }^{m}\left(z\right)/\partial \nu \right]}_{\nu =n}$ with respect to the degree and the partial derivative ${\left[\partial {P}_{n}^{\mu }\left(z\right)/\partial \mu \right]}_{\mu =m}$ with respect to the order for $m,n\in {ℕ}_{0}$. This relationship is used to derive some new representations for ${\left[\partial {P}_{\nu }^{m}\left(z\right)/\partial \nu \right]}_{\nu =n}$ based on representations for ${\left[\partial {P}_{n}^{\mu }\left(z\right)/\partial \mu \right]}_{\mu =m}$ obtained earlier by the author in [J. Math. Chem. 46, No. 1, 231–260 (2009; Zbl 05571791)] for integer values of $m$ and $n$ with $0\le m\le n$. Moreover, several new expressions are derived for the associated Legendre functions ${Q}_{n}^{m}\left(z\right)$ of the second kind of integer degree and integer order.
##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 33C45 Orthogonal polynomials and functions of hypergeometric type
DLMF