Solutions to the associated Laguerre differential equation with and an integer are called associated Laguerre polynomials
(Arfken 1985, p. 726) or, in older literature, Sonine polynomials (Sonine 1880,
p. 41; Whittaker and Watson 1990, p. 352). Associated Laguerre polynomials
are implemented in the Wolfram Language
as LaguerreL[n,
k, x]. In terms of the unassociated Laguerre
polynomials,
where the usual factor of in the denominator has been suppressed (Roman 1984, p. 31).
Many interesting properties of the associated Laguerre polynomials follow from the
fact that (Roman 1984, p. 31).
The associated Laguerre polynomials are given explicitly by the formula
A generalization of the associated Laguerre polynomial to not necessarily an integer is called a Laguerre function (Arfken
1985, p. 726) or a generalized Laguerre function (Abramowitz and Stegun 1972,
p. 775). These generalized Laguerre polynomial can be defined as
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