Abstract
In our recent works (R. Szmytkowski, J. Phys. A 39:15147, 2006; corrigendum: 40:7819, 2007; addendum: 40:14887, 2007), we have investigated the derivative of the Legendre function of the first kind, P ν(z), with respect to its degree ν. 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 ν, 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 \({[\partial P_{\nu}^{-m}(\pm z)/\partial\nu]_{\nu=n}}\). The present paper is a complementary to a recent one (R. Szmytkowski, J. Math. Chem 46:231, 2009), in which the derivative \({\partial P_{n}^{\mu}(z)/\partial\mu}\) has been investigated.
Article PDF
Similar content being viewed by others
References
Szmytkowski R.: On the derivative of the Legendre function of the first kind with respect to its degree. J. Phys. A 39, 15147 (2006) [corrigendum: 40, 7819 (2007)]
Jolliffe A.E.: A form for \({\frac{\mathrm{d}}{\mathrm{d}n}P_{n}(\mu)}\), where P n (μ) is the Legendre polynomial of degree n. Mess. Math. 49, 125 (1919)
I’A Bromwich T.J.: Certain potential functions and a new solution of Laplace’s equation. Proc. Lond. Math. Soc. 12, 100 (1913)
Schelkunoff S.A.: Theory of antennas of arbitrary size and shape. Proc. IRE 29, 493 (1941) [corrigendum: 31, 38 (1943); reprint: Proc. IEEE 72, 1165 (1984)]
Magnus W., Oberhettinger F., Soni R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edn. Springer, Berlin (1966)
Szmytkowski R.: Addendum to ‘On the derivative of the Legendre function of the first kind with respect to its degree’. J. Phys. A 40, 14887 (2007)
Carslaw H.S.: Integral equations and the determination of Green’s functions in the theory of potential. Proc. Edinburgh Math. Soc. 31, 71 (1913)
Carslaw H.S.: The scattering of sound waves by a cone. Math. Ann. 75, 133 (1914) [corrigendum: 75, 592 (1914)]
Carslaw H.S.: The Green’s function for the equation \({{\nabla}^{2}u + k^{2}u = 0}\). Proc. Lond. Math. Soc. 13, 236 (1914)
Macdonald H.M.: A class of diffraction problems. Proc. Lond. Math. Soc. 14, 410 (1915)
Carslaw H.S.: Introduction to the Mathematical Theory of the Conduction of Heat in Solids, pp. 145–147. Macmillan, London (1921)
H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1947) pp. 214 and 318
Smythe W.R.: Static and Dynamic Electricity, 2nd edn, pp. 156–157. McGraw-Hill, New York (1950)
Smythe W.R.: Static and Dynamic Electricity, 3rd edn, pp. 166–167. McGraw-Hill, New York (1968)
Felsen L.B.: Backscattering from wide-angle and narrow-angle cones. J. Appl. Phys. 26, 138 (1955)
Bailin L.L., Silver S.: Exterior electromagnetic boundary value problems for spheres and cones. IRE Trans. Antennas Propag. 4, 5 (1956) [corrigendum: 5, 313 (1957)]
Felsen L.B.: Plane-wave scattering by small-angle cones. IRE Trans. Antennas Propag. 5, 121 (1957)
Felsen L.B.: Radiation from ring sources in the presence of a semi-infinite cone. IRE Trans. Antennas Propag. 7, 168 (1959) [corrigendum: 7, 251 (1959)]
Jones D.S.: The Theory of Electromagnetism, pp. 614. Pergamon, Oxford (1964)
Bowman J.J.: Electromagnetic and Acoustic Scattering by Simple Shapes. In: Bowman, J.J., Senior, T.B.A., Uslenghi, P.L.E. (eds) , pp. 637. North-Holland, Amsterdam (1969)
Felsen L.B., Marcuvitz N.: Radiation and Scattering of Waves. Prentice-Hall, Englewood Cliffs, NJ (1973) [reprinted: IEEE Press, Piscataway, NJ, 1994], pp. 320, 321, 703 and 734
Galitsyn A.S., Zhukovskii A.N.: Integral Transforms and Special Functions in Heat Conduction Problems. Naukova Dumka, Kiev (1976) (in Russian), pp. 236, 237 and 239
Ariyasu J.C., Mills D.L.: Inelastic electron scattering by long-wavelength, acoustic phonons; image potential modulation as a mechanism. Surf. Sci. 155, 607 (1985) (appendix B)
Jones D.S.: Acoustic and Electromagnetic Waves, pp. 591. Clarendon, Oxford (1986)
Bauer H.F.: Mass transport in a three-dimensional diffusor or confusor. Wärme-Stoffübertrag 21, 51 (1987)
Bauer H.F.: Response of axially excited spherical and conical liquid systems with anchored edges. Forsch. Ing.-Wes. 58(4), 96 (1992)
Broadbent E.G., Moore D.W.: The inclination of a hollow vortex with an inclined plane and the acoustic radiation produced. Proc. R. Soc. Lond. A 455, 1979 (1999)
Van Bladel J.: Electromagnetic Fields, 2nd edn. IEEE Press, Piscataway (2007) (Section 16.7.1)
Szmytkowski R.: The Green’s function for the wavized Maxwell fish-eye problem. J. Phys. A 44, 065203 (2011)
R. Szmytkowski, Some differentiation formulas for Legendre polynomials, arXiv:0910.4715
Hobson E.W.: The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, Cambridge (1931) [reprinted: Chelsea, New York, 1955]
Robin L.: Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vol. 1. Gauthier-Villars, Paris (1957)
Robin L.: Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vol. 2. Gauthier-Villars, Paris (1958)
Robin L.: Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vol. 3. Gauthier-Villars, Paris (1959)
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, 231 (2009)
R. Szmytkowski, 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
Robin L.: Derivée de la fonction associée de Legendre, de première espèce, par rapport à son degré. Compt. Rend. Acad. Sci. Paris 242, 57 (1956)
Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products, 5th edn. Academic, San Diego (1994)
Prudnikov A.P., Brychkov Yu.A., Marichev O.I.: Integrals and Series. Special Functions. Supplementary Chapters, 2nd edn. Fizmatlit, Moscow (2003) (in Russian)
Hostler L.: Nonrelativistic Coulomb Green’s function in momentum space. J. Math. Phys. 5, 1235 (1964)
Brychkov Yu.A.: On the derivatives of the Legendre functions \({P_{\nu}^{\mu}(z)}\) and \({Q_{\nu}^{\mu}(z)}\) with respect to μ and ν. Integral Transforms Spec. Funct. 21, 175 (2010)
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, 581 (2010)
Brychkov Yu.A.: Handbook of Special Functions. Derivatives, Integrals, Series and Other Formulas. Chapman & Hall/CRC, Boca Raton, FL (2008)
Magnus W., Oberhettinger F.: Formeln und Sätze für die speziellen Funktionen der mathematischen Physik, 2nd edn. Springer, Berlin (1948)
Stegun I.A.: Handbook of Mathematical Functions. In: Abramowitz, M., Stegun, I.A. (eds) , pp. 331. Dover, New York (1965)
Tsu R.: The evaluation of incomplete normalization integrals and derivatives with respect to the order of associated Legendre polynomials. J. Math. Phys. 40, 232 (1961)
Carlson B.C.: Dirichlet averages of x t log x. SIAM J. Math. Anal. 18, 550 (1987)
Schendel L.: Zusatz zu der Abhandlung über Kugelfunctionen S. 86 des 80. Bandes. J. Reine Angew. Math. (Borchardt J.) 82, 158 (1877)
Snow Ch.: Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd edn. National Bureau of Standards, Washington, DC (1952)
Szmytkowski R.: Closed form of the generalized Green’s function for the Helmholtz operator on the two-dimensional unit sphere. J. Math. Phys. 47, 063506 (2006)
Szegö G.: Orthogonal Polynomials. American Mathematical Society, New York (1939) (chapter 4)
Fröhlich J.: Parameter derivatives of the Jacobi polynomials and the Gaussian hypergeometric function. Integral Transforms Spec. Funct. 2, 253 (1994)
R. Szmytkowski, A note on parameter derivatives of classical orthogonal polynomials. arXiv:0901.2639
Acknowledgments
The author wishes to thank an anonymous referee to Ref. [50], whose suggestion to use the contour-integration technique to evaluate the derivative [∂ Pν(z)/∂ν] ν=n inspired the present work.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution,and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
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 49, 1436–1477 (2011). https://doi.org/10.1007/s10910-011-9826-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-011-9826-3