Sains Ma1aysiana 26(1): 69-77 (1997)                                                                           Sains Fizis dan Gunaan/

                                                                                                                                         Physical and Applied Sciences

 

Laguerre Polynomials and Functions

 

 

Karim B. Yaacob

Faculty of Science and Art

Universiti Putra Malaysia Terengganu

21030 Kuala Terengganu Terengganu Malaysia

 

 

ABSTRAK

 

Mengikut teori persamaan perbezaan hipergeometrik konfluen, polinomial Laguerre Lnm(x), memenuhi persamaan xy" + (m+l-x)y' + (n-m)y=0, untuk kesemua nombor genap positif n m, termasuk sifar. Di dalam keadaan n bukan nombor genap, penyelesaian kepada persamaan Laguerre bukan berbentuk polinomial, tetapi suatu siri tak terhingga yang sukar untuk disimpulkan di dalam bentuk fungsi tertutup. Suatu kaedah baru dikenali di mana jadual polinomial Laguerre (n nombor genap) dan fungsi Laguerre (n bukan nombor genap) dapat diterbitkan melalui penyelesaian kepada persamaan kamiran. Kaedah ini menggunakan teknik novel melalui penyelesaian kepada persamaan Bateman dan persamaan Yaacob-Bateman untuk mendapat penyelesaian kepada persamaan jejari Schroedinger untuk atom hidrogen.

 

ABSTRACT

 

In the theory of the confluent hypergeometric differential equation, Laguerre polynomials Lnm(x), satisfy the equation xy" + (m+l-x)y' + (n-m)y=0, for non-negative integers of n m, including zero. When n are non-integers, the solutions to the equation are not polynomials, but infinite series that is difficult to evaluate in closed from. A new method is introduced whereby it is possible to generate a table of Laguerre polynomials (for integer n) and Laguerre functions (for non-integer n) through evaluation of one common integral equation. The method uses the Bateman and the Yaacob-Bateman equations as new tools of solution to the radial Schroedinger equation for the hydrogen atom.

 

 

RUJUKAN/REFERENCES

 

Bateman, H. 1931. The k-function, a particular case of the confluent hypergeometric function. Trans. Am. Math. Soc. 33: 817.

Whittaker, E.T. 1903-04. An expression of certain known functions as generalized hypergeometric functions. Bul. Am. Math. Soc. 10: 132.

Shabde, N.G. 1937. On some kn-function formulate. J. Indian Math. Soc. 2: 27.

Chakrabarty, N.K. 1953. On a generalisation of Bateman's k-function. Bull. Cal. Math. Soc. 45: 1

Srivastava, T.N. 1974. On the generalised Bateman k-function. Bull, Cal Math. Soc. 66: 185.

Yaacob, K.B. 1988. The hydrogen atom and Bateman functions. Sing. J. Physics. 5: 77.

Kummer, E.E. 1837. De integralibus quibusdam definitis et seriebus infinitis. J. Math. 17: 228; 1840. ibid. 20:1.

Bateman, H. 1931. Solutions of a certain partial differential equation. Proc. N.A.S. 17: 562.

Yaacob, K.B. 1987. An integral transform solution for the one-dimentsional hydrogen atom. Sains Malaysiana 16: 193.

Yaacob, K.B. 1988. An integral representation solution for the one-dimensional harmonic oscillator. Sains Malaysiana 17: 233.

Yaacob, K.B. 1989. Note on the solution to the confluent hypergeometric differential equation zy" + my' + (n-y)y = 0. Sing. J. Phys. 6: 39.

Yaacob, K.B. 1990. Extended plane polar coordinate systems using the two-dimensional rotation operator. Sing. J. Phys. 7: 25.

Grashteyn, I.S. & Ryzhik. 1980. Table of Integrals, Series and Products. New York: Academic Press.

 

 

 

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