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 SIGMA, 2011, Volume 7, 050, 16 pp. (Mi sigma608)

On Parameter Differentiation for Integral Representations of Associated Legendre Functions

Howard S. Cohlab

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

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 $f:\mathbb C\setminus\{-1,1\}\to\mathbb C$ given by $f(z)=z/(\sqrt{z+1}\sqrt{z-1})$.

Keywords: Legendre functions; modified Bessel functions; derivatives

DOI: https://doi.org/10.3842/SIGMA.2011.050

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ArXiv: 1101.3756
MSC: 31B05; 31B10; 33B10; 33B15; 33C05; 33C10
Received: January 19, 2011; in final form May 4, 2011; Published online May 24, 2011
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Citation: Howard S. Cohl, “On Parameter Differentiation for Integral Representations of Associated Legendre Functions”, SIGMA, 7 (2011), 050, 16 pp.

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Szmytkowski R., “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)”, J Math Anal Appl, 386:1 (2012), 332–342
2. Szmytkowski R., “On the Derivatives Partial Derivative P-2(Nu)(Z)/Partial Derivative Nu(2) and Partial Derivative Q(Nu)(Z)/Partial Derivative Nu of the Legendre Functions With Respect to Their Degrees”, Integral Transform. Spec. Funct., 28:9 (2017), 645–662
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