Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the derivative of the associated Legendre function of the first kind
  of integer order with respect to its degree   In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum:
40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivative
of the Legendre function of the first kind, $P_{\nu}(z)$, with respect to its
degree $\nu$. In the present work, we extend these studies and construct
several representations of the derivative of the associated Legendre function
of the first kind, $P_{\nu}^{\pm m}(z)$, with respect to the degree $\nu$, for
$m\in\mathbb{N}$. At first, we establish several contour-integral
representations of $\partial P_{\nu}^{\pm m}(z)/\partial\nu$. They are then
used to derive Rodrigues-type formulas for $[\partial P_{\nu}^{\pm
m}(z)/\partial\nu]_{\nu=n}$ with $n\in\mathbb{N}$. Next, some closed-form
expressions for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ are
obtained. These results are applied to find several representations, both
explicit and of the Rodrigues type, for the associated Legendre function of the
second kind of integer degree and order, $Q_{n}^{\pm m}(z)$; the explicit
representations are suitable for use for numerical purposes in various regions
of the complex $z$-plane. Finally, the derivatives
$[\partial^{2}P_{\nu}^{m}(z)/\partial\nu^{2}]_{\nu=n}$, $[\partial
Q_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$ and $[\partial
Q_{\nu}^{m}(z)/\partial\nu]_{\nu=-n-1}$, all with $m>n$, are evaluated in terms
of $[\partial P_{\nu}^{-m}(\pm z)/\partial\nu]_{\nu=n}$.

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The categories of this paper are: hep-ph
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: q-fin
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The categories of this paper are: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: dg-ga
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The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-th
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The categories of this paper are: eess
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The categories of this paper are: acc-phys