
SIGMA 7 (2011), 050, 16 pages arXiv:1101.3756
https://doi.org/10.3842/SIGMA.2011.050
Contribution to the Special Issue “Symmetry, Separation, Superintegrability and Special Functions (S^{4})”
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
oddhalfinteger degrees, for general complexorders, and derivatives
with respect to the order are evaluated at integerorders,
for general complexdegrees. 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.
pdf (496 kb)
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References
 Abramowitz M., Stegun I.A.,
Handbook of mathematical functions with formulas, graphs, and mathematical tables,
National Bureau of Standards Applied Mathematics Series, Vol. 55, U.S. Government Printing Office, Washington, D.C., 1964.
 Apelblat A., Kravitsky N.,
Integral representations of derivatives and integrals with respect to the order of the Bessel functions J_{ν}(t), I_{ν}(t), the Anger function J_{ν}(t) and the integral Bessel function Ji_{ν}(t),
IMA J. Appl. Math. 34 (1985), 187210.
 Brychkov Yu.A.,
Handbook of special functions: derivatives, integrals, series and other formulas,
CRC Press, Boca Raton, FL, 2008.
 Brychkov Yu.A.,
On the derivatives of the Legendre functions P_{ν}^{μ}(z) and Q_{ν}^{μ}(z) with respect to μ and ν,
Integral Transforms Spec. Funct. 21 (2010), 175181.
 Cohl H.S.,
Derivatives with respect to the degree and order of associated Legendre functions for z>1 using modified Bessel functions,
Integral Transforms Spec. Funct. 21 (2010), 581588,
arXiv:0911.5266.
 Cohl H.S., Rau A.R.P., Tohline J.E., Browne D.A., Cazes J.E., Barnes E.I.,
Useful alternative to the multipole expansion of 1/r potentials,
Phys. Rev. A 64 (2001), 052509, 5 pages.
 Cohl H.S., Tohline J.E.,
A compact cylindrical Green's function expansion for the solution of potential problems,
Astrophys. J. 527 (1999), 86101.
 Cohl H.S., Tohline J.E., Rau A.R.P., Srivastava H. M.,
Developments in determining the gravitational potential using toroidal functions,
Astronom. Nachrichten 321 (2000), 363372.
 Gradshteyn I.S., Ryzhik I.M.,
Table of integrals, series, and products, 7th ed.,
Elsevier/Academic Press, Amsterdam, 2007.
 Hobson E.W.,
The theory of spherical and ellipsoidal harmonics,
Chelsea Publishing Company, New York, 1955.
 Lang S.,
Real and functional analysis, 3rd ed.,
Graduate Texts in Mathematics, Vol. 142,
SpringerVerlag, New York, 1993.
 Magnus W., Oberhettinger F., Soni R.P.,
Formulas and theorems for the special functions of mathematical physics,
3rd ed., Die Grundlehren der mathematischen Wissenschaften, Band 52, SpringerVerlag, New York, 1966.
 Olver F.W.J.,
Asymptotics and special functions,
AKP Classics, A K Peters Ltd., Wellesley, MA, 1997.
 Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors),
NIST handbook of mathematical functions,
U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, 2010.
 Prudnikov A.P., Brychkov Yu.A., Marichev O.I.,
Integrals and series, Vol. 2, Special functions, 2nd ed.,
Gordon & Breach Science Publishers, New York, 1988.
 Silverman R.A.,
Complex analysis with applications,
PrenticeHall Inc., Englewood Cliffs, N.J., 1974.
 Szmytkowski R.,
On the derivative of the Legendre function of the first kind with respect to its degree,
J. Phys. A: Math. Gen. 39 (2006), 1514715172,
Addendum, J. Phys. A: Math. Theor. 40 (2007), 1488714891,
Corrigendum, J. Phys. A: Math. Theor. 40 (2007), 78197820,
arXiv:0907.3217.
 Szmytkowski R.,
A note on parameter derivatives of classical orthogonal polynomials,
arXiv:0901.2639.
 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),
arXiv:0910.4550.
 Szmytkowski R.,
On the derivative of the associated Legendre function of the first
kind of integer degree with respect to its order (with applications to the
construction of the associated Legendre function of the second kind of
integer degree and order),
J. Math. Chem. 46 (2009), 231260.
 Szmytkowski R.,
On the derivative of the associated Legendre function of the first
kind of integer order with respect to its degree (with applications to the
construction of the associated Legendre function of the second kind of
integer degree and order),
J. Math. Chem., to appear,
arXiv:0907.3217.
 Szmytkowski R.,
Green's function for the wavized Maxwell fisheye problem,
J. Phys. A: Math. Theor. 44 (2011), 065203, 14 pages.
 Watson G.N.,
A treatise on the theory of Bessel functions, 2nd ed.,
Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1944.
 Whipple F.J.W.,
A symmetrical relation between Legendre's functions with parameters coshα and cothα,
Proc. London Math. Soc. 16 (1917), 301314.

