Journal of Mathematical Chemistry

, Volume 49, Issue 7, pp 1436–1477 | Cite as

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)

Open Access
Original Paper

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, Pν(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.

Keywords

Special functions Legendre functions Spherical harmonics Parameter derivatives 

Mathematics Subject Classification (2000)

33C45 33C05 

References

  1. 1.
    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)]CrossRefGoogle Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    I’A Bromwich T.J.: Certain potential functions and a new solution of Laplace’s equation. Proc. Lond. Math. Soc. 12, 100 (1913)CrossRefGoogle Scholar
  4. 4.
    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)]CrossRefGoogle Scholar
  5. 5.
    Magnus W., Oberhettinger F., Soni R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edn. Springer, Berlin (1966)Google Scholar
  6. 6.
    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)CrossRefGoogle Scholar
  7. 7.
    Carslaw H.S.: Integral equations and the determination of Green’s functions in the theory of potential. Proc. Edinburgh Math. Soc. 31, 71 (1913)CrossRefGoogle Scholar
  8. 8.
    Carslaw H.S.: The scattering of sound waves by a cone. Math. Ann. 75, 133 (1914) [corrigendum: 75, 592 (1914)]CrossRefGoogle Scholar
  9. 9.
    Carslaw H.S.: The Green’s function for the equation \({{\nabla}^{2}u + k^{2}u = 0}\). Proc. Lond. Math. Soc. 13, 236 (1914)CrossRefGoogle Scholar
  10. 10.
    Macdonald H.M.: A class of diffraction problems. Proc. Lond. Math. Soc. 14, 410 (1915)Google Scholar
  11. 11.
    Carslaw H.S.: Introduction to the Mathematical Theory of the Conduction of Heat in Solids, pp. 145–147. Macmillan, London (1921)Google Scholar
  12. 12.
    H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids (Clarendon, Oxford, 1947) pp. 214 and 318Google Scholar
  13. 13.
    Smythe W.R.: Static and Dynamic Electricity, 2nd edn, pp. 156–157. McGraw-Hill, New York (1950)Google Scholar
  14. 14.
    Smythe W.R.: Static and Dynamic Electricity, 3rd edn, pp. 166–167. McGraw-Hill, New York (1968)Google Scholar
  15. 15.
    Felsen L.B.: Backscattering from wide-angle and narrow-angle cones. J. Appl. Phys. 26, 138 (1955)CrossRefGoogle Scholar
  16. 16.
    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)]Google Scholar
  17. 17.
    Felsen L.B.: Plane-wave scattering by small-angle cones. IRE Trans. Antennas Propag. 5, 121 (1957)CrossRefGoogle Scholar
  18. 18.
    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)]CrossRefGoogle Scholar
  19. 19.
    Jones D.S.: The Theory of Electromagnetism, pp. 614. Pergamon, Oxford (1964)Google Scholar
  20. 20.
    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)Google Scholar
  21. 21.
    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 734Google Scholar
  22. 22.
    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 239Google Scholar
  23. 23.
    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)CrossRefGoogle Scholar
  24. 24.
    Jones D.S.: Acoustic and Electromagnetic Waves, pp. 591. Clarendon, Oxford (1986)Google Scholar
  25. 25.
    Bauer H.F.: Mass transport in a three-dimensional diffusor or confusor. Wärme-Stoffübertrag 21, 51 (1987)CrossRefGoogle Scholar
  26. 26.
    Bauer H.F.: Response of axially excited spherical and conical liquid systems with anchored edges. Forsch. Ing.-Wes. 58(4), 96 (1992)CrossRefGoogle Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    Van Bladel J.: Electromagnetic Fields, 2nd edn. IEEE Press, Piscataway (2007) (Section 16.7.1)CrossRefGoogle Scholar
  29. 29.
    Szmytkowski R.: The Green’s function for the wavized Maxwell fish-eye problem. J. Phys. A 44, 065203 (2011)CrossRefGoogle Scholar
  30. 30.
    R. Szmytkowski, Some differentiation formulas for Legendre polynomials, arXiv:0910.4715Google Scholar
  31. 31.
    Hobson E.W.: The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, Cambridge (1931) [reprinted: Chelsea, New York, 1955]Google Scholar
  32. 32.
    Robin L.: Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vol. 1. Gauthier-Villars, Paris (1957)Google Scholar
  33. 33.
    Robin L.: Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vol. 2. Gauthier-Villars, Paris (1958)Google Scholar
  34. 34.
    Robin L.: Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vol. 3. Gauthier-Villars, Paris (1959)Google Scholar
  35. 35.
    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)CrossRefGoogle Scholar
  36. 36.
    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:0910.4550Google Scholar
  37. 37.
    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)Google Scholar
  38. 38.
    Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products, 5th edn. Academic, San Diego (1994)Google Scholar
  39. 39.
    Prudnikov A.P., Brychkov Yu.A., Marichev O.I.: Integrals and Series. Special Functions. Supplementary Chapters, 2nd edn. Fizmatlit, Moscow (2003) (in Russian)Google Scholar
  40. 40.
    Hostler L.: Nonrelativistic Coulomb Green’s function in momentum space. J. Math. Phys. 5, 1235 (1964)CrossRefGoogle Scholar
  41. 41.
    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)CrossRefGoogle Scholar
  42. 42.
    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)CrossRefGoogle Scholar
  43. 43.
    Brychkov Yu.A.: Handbook of Special Functions. Derivatives, Integrals, Series and Other Formulas. Chapman & Hall/CRC, Boca Raton, FL (2008)Google Scholar
  44. 44.
    Magnus W., Oberhettinger F.: Formeln und Sätze für die speziellen Funktionen der mathematischen Physik, 2nd edn. Springer, Berlin (1948)Google Scholar
  45. 45.
    Stegun I.A.: Handbook of Mathematical Functions. In: Abramowitz, M., Stegun, I.A. (eds) , pp. 331. Dover, New York (1965)Google Scholar
  46. 46.
    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)Google Scholar
  47. 47.
    Carlson B.C.: Dirichlet averages of x t log x. SIAM J. Math. Anal. 18, 550 (1987)CrossRefGoogle Scholar
  48. 48.
    Schendel L.: Zusatz zu der Abhandlung über Kugelfunctionen S. 86 des 80. Bandes. J. Reine Angew. Math. (Borchardt J.) 82, 158 (1877)CrossRefGoogle Scholar
  49. 49.
    Snow Ch.: Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd edn. National Bureau of Standards, Washington, DC (1952)Google Scholar
  50. 50.
    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)CrossRefGoogle Scholar
  51. 51.
    Szegö G.: Orthogonal Polynomials. American Mathematical Society, New York (1939) (chapter 4)Google Scholar
  52. 52.
    Fröhlich J.: Parameter derivatives of the Jacobi polynomials and the Gaussian hypergeometric function. Integral Transforms Spec. Funct. 2, 253 (1994)CrossRefGoogle Scholar
  53. 53.
    R. Szmytkowski, A note on parameter derivatives of classical orthogonal polynomials. arXiv:0901.2639Google Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Atomic Physics Division, Department of Atomic Physics and Luminescence, Faculty of Applied Physics and MathematicsGdańsk University of TechnologyGdańskPoland

Personalised recommendations