### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 050, 16 pages      arXiv:1101.3756      https://doi.org/10.3842/SIGMA.2011.050
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

### On Parameter Differentiation for Integral Representations of Associated Legendre Functions

Howard S. Cohl a, b
a) Applied and Computational Mathematics Division, Information Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland, USA
b) Department of Mathematics, University of Auckland, 38 Princes Str., Auckland, New Zealand

Received January 19, 2011, in final form May 04, 2011; Published online May 24, 2011

Abstract
For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for associated Legendre functions of the first and second kind with respect to the degree are evaluated at odd-half-integer degrees, for general complex-orders, and derivatives with respect to the order are evaluated at integer-orders, for general complex-degrees. We also discuss the properties of the complex function f: C\{−1,1}→C given by f(z)=z/((z+1)1/2(z−1)1/2).

Key words: Legendre functions; modified Bessel functions; derivatives.

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