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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). (English) Zbl 05571791
Summary: The derivative of the associated Legendre function of the first kind of integer degree with respect to its order, P n μ (z)/μ, is studied. After deriving and investigating general formulas for μ arbitrary complex, a detailed discussion of [P n μ (z)/μ] μ=±m , where m is a non-negative integer, is carried out. The results are applied to obtain several explicit expressions for the associated Legendre function of the second kind of integer degree and order, Q n ±m (z). In particular, we arrive at formulas which generalize to the case of Q n ±m (z)(0mn) the well-known Christoffel’s representation of the Legendre function of the second kind, Q n (z). The derivatives [ 2 P n μ (z)/μ 2 ] μ=m ,[Q n μ (z)/μ] μ=m and [Q -n-1 μ (z)/μ] μ=m , all with m>n, are also evaluated.
MSC:
92EChemistry
References:
[1]Todhunter I.: An Elementary Treatise on Laplace’s Functions, Lamé’s Functions and Bessel’s Functions. Macmillan, London (1975)
[2]Ferrers N.M.: An Elementary Treatise on Spherical Harmonics. Macmillan, London (1877)
[3]Neumann F.: Beiträge zur Theorie der Kugelfunctionen. Teubner, Leipzig (1878)
[4]Heine E.: Handbuch der Kugelfunctionen, vol. 1, 2nd edn. Reimer, Berlin (1878)
[5]Heine E.: Handbuch der Kugelfunctionen, vol. 2, 2nd edn. Reimer, Berlin (1881)
[6]Olbricht R.: Nova Acta Leop. Carol. Akad. 52, 1 (1887)
[7]W.E. Byerly, An Elementary Treatise on Fourier’s Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics (Ginn, Boston, 1893) [reprinted: (Dover, Mineola, NY, 2003)]
[8]Hobson E.W.: Philos. Trans. R. Soc. Lond. A 187, 443 (1896) · doi:10.1098/rsta.1896.0013
[9]A. Wangerin, Theorie der Kugelfunktionen und der verwandten Funktionen, insbesondere der Lamé’schen und Bessel’schen, in Encyklopädie der mathematischen Wissenschaften, vol. 2.1 (Teubner, Leipzig, 1904), p. 695
[10]Barnes E.W.: Q. J. Pure Appl Math. 39, 97 (1907)
[11]Wangerin A.: Theorie des Potentials und der Kugelfunktionen, vol. 2. de Gruyter, Berlin (1921)
[12]Hobson E.W.: The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press, Cambridge (1931) [reprinted: (Chelsea, New York, 1955)]
[13]Ch. Snow, Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory, 2nd edn. (National Bureau of Standards, Washington, DC, 1952)
[14]A. Erdélyi (ed.), Higher Transcendental Functions, vol. 1 (McGraw-Hill, New York, 1953), Chap. III
[15]Morse P.M., Feshbach H.: Methods of Theoretical Physics. McGraw-Hill, New York (1953)
[16]Lense J.: Kugelfunktionen, 2nd edn. Geest & Portig, Leipzig (1954)
[17]L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vol. 1 (Gauthier-Villars, Paris, 1957)
[18]L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vol. 2 (Gauthier-Villars, Paris, 1958)
[19]L. Robin, Fonctions Sphériques de Legendre et Fonctions Sphéroïdales, vol. 3 (Gauthier-Villars, Paris, 1959)
[20]E. Jahnke, F. Emde, F. Lösch, Tafeln höherer Funktionen, 6th edn. (Teubner, Stuttgart, 1960)
[21]A. Kratzer, W. Franz, Transzendente Funktionen (Akademische Verlagsgesellschaft, Leipzig, 1960), Chap. 5
[22]I.A. Stegun, in Legendre functions, ed. by M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions (Dover, New York, 1965), p. 331
[23]Magnus W., Oberhettinger F., Soni R.P.: Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd edn. Springer, Berlin (1966)
[24]MacRobert T.M.: Spherical Harmonics, 3rd edn. Pergamon, Oxford (1967)
[25]Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products, 5th edn. Academic, San Diego (1994)
[26]N.M. Temme, Special Functions. An Introduction to the Classical Functions of Mathematical Physics (Wiley, New York, 1996), Chap. 8
[27]A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series. Special Functions (Nauka, Moscow, 1983) (in Russian)
[28]A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series. Special Functions. Supplementary Chapters, 2nd edn. (Fizmatlit, Moscow, 2003) (in Russian)
[29]Yu.A. Brychkov, Special Functions. Derivatives, Integrals, Series, and Other Formulas (Fizmatlit, Moscow, 2006) (in Russian)
[30]F.E. Neumann, J. Reine Angew. Math. (Crelle J.) 37, 21 (1848) [reprinted in: Franz Neumanns gesammelte Werke, vol. 3 (Teubner, Leipzig, 1912), p. 439]
[31]Sugiura Y.: Z. Phys. 45, 484 (1927) · doi:10.1007/BF01329207
[32]Kemble E.C., Zener C.: Phys. Rev. 33, 512 (1929) · doi:10.1103/PhysRev.33.512
[33]J.C. Slater, Quantum Theory of Molecules and Solids, vol. 1. Electronic Structure of Molecules (McGraw-Hill, New York, 1963), Appendix 6
[34]Yasui J., Saika A.: J. Chem. Phys. 76, 468 (1982) · doi:10.1063/1.442745
[35]J. Hinze, F. Biegler-König, in Self-consistent field. Theory and Applications, ed. by R. Carbó, M. Klobukowski (Elsevier, Amsterdam, 1990), p. 405
[36]Harris F.E.: Int. J. Quant. Chem. 88, 701 (2002) · doi:10.1002/qua.10181
[37]Vanne Y.V., Saenz A.: J. Phys. B 37, 4101 (2004) · doi:10.1088/0953-4075/37/20/005
[38]Takagi H., Nakamura H.: J. Phys. B 13, 2619 (1980) · doi:10.1088/0022-3700/13/13/020
[39]El-Aasser M.A., Abdel-Raouf M.A.: J. Phys. B 40, 1801 (2007) · doi:10.1088/0953-4075/40/10/015
[40]P.J. Davis, in Gamma function and related functions, ed. by M. Abramowitz, I.A. Stegun. Handbook of Mathematical Functions (Dover, New York, 1965), p. 253
[41]Brown G.J.N.: J. Phys. A 28, 2297 (1995) · Zbl 0860.33008 · doi:10.1088/0305-4470/28/8/020
[42]Watson G.N.: Proc. Lond. Math. Soc. 17, 241 (1918) · doi:10.1112/plms/s2-17.1.241