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2018•Unpublished venueRequires access

Laguerre Polynomials

Selçuk Ş. Bayın

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Abstract

Laguerre polynomials, Ln(x), are named after the French mathematician Edmond Laguerre. They are the solutions of the Laguerre equation given in this chapter. The chapter encounters Laguerre polynomials in quantum mechanics in the solutions of the hydrogen atom problem. Using the method that is used in the chapter for the Legendre polynomials, two recursion relations is obtained for the Laguerre polynomials. Compared with the Legendre polynomials, it is possible that the Laguerre polynomials are orthogonal with respect to the weight function e-x. One of the definitions of the Laguerre polynomials is given in terms of the Rodriguez formula. To show the equivalence of this formula with the other definitions, the Leibniz formula is used in the chapter.

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Laguerre polynomials, Ln(x), are named after the French mathematician Edmond Laguerre. They are the solutions of the Laguerre equation given in this chapter. The chapter encounters Laguerre polynomials in quantum mechanics in the solutions of the hydrogen atom problem. Using the method that is used in the chapter for the Legendre polynomials, two recursion relations is obtained for the Laguerre polynomials. Compared with the Legendre polynomials, it is possible that the Laguerre polynomials are orthogonal with respect to the weight function e-x. One of the definitions of the Laguerre polynomials is given in terms of the Rodriguez formula. To show the equivalence of this formula with the other definitions, the Leibniz formula is used in the chapter.

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Available abstract

Laguerre polynomials, Ln(x), are named after the French mathematician Edmond Laguerre. They are the solutions of the Laguerre equation given in this chapter. The chapter encounters Laguerre polynomials in quantum mechanics in the solutions of the hydrogen atom problem. Using the method that is used in the chapter for the Legendre polynomials, two recursion relations is obtained for the Laguerre polynomials. Compared with the Legendre polynomials, it is possible that the Laguerre polynomials are orthogonal with respect to the weight function e-x. One of the definitions of the Laguerre polynomials is given in terms of the Rodriguez formula. To show the equivalence of this formula with the other definitions, the Leibniz formula is used in the chapter.

Key concepts: Laguerre polynomials, Laguerre's method, Classical orthogonal polynomials, Mathematics, Orthogonal polynomials, Legendre polynomials, Discrete orthogonal polynomials, Gegenbauer polynomials

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