26 research outputs found
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, , with respect to its
degree . In the present work, we extend these studies and construct
several representations of the derivative of the associated Legendre function
of the first kind, , with respect to the degree , for
. At first, we establish several contour-integral
representations of . They are then
used to derive Rodrigues-type formulas for with . Next, some closed-form
expressions for 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, ; the explicit
representations are suitable for use for numerical purposes in various regions
of the complex -plane. Finally, the derivatives
, and , all with , are evaluated in terms
of .Comment: LateX, 40 pages, 1 figure, extensive referencin