Where, " } } is the associated Laguerre polynomial of order and index, and is the spot size of the mode corresponding to the Gaussian beam radius.
12.
When " a " is a non-positive integer then Kummer's function ( if it is defined ) is a ( generalized ) Laguerre polynomial.
13.
The wave functions of an electron in a generalized Laguerre polynomials ( these are defined differently by different authors & mdash; see main article on them and the hydrogen atom ).
14.
:I have no constructive response to your hypothetical Laguerre polynomial example because there is no such article and I don't think it is useful to discuss these matters in the abstract.
15.
In mathematics, "'Angelescu polynomials "'? " n " ( " x " ) are generalizations of the Laguerre polynomials introduced by given by the generating function
16.
The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the Gegenbauer polynomials, the Chebyshev polynomials, and the Legendre polynomials.
17.
But the above answer for the case " r " = 2 gives an orthogonality relation, whence the " P " " n "'s are the Laguerre polynomials ( up to a sign that is easily decided ).
18.
Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of " n " ! than the definition used here . ( Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials .)
19.
Physicists sometimes use a definition for the Laguerre polynomials which is larger by a factor of " n " ! than the definition used here . ( Likewise, some physicists may use somewhat different definitions of the so-called associated Laguerre polynomials .)
20.
I don't see a claim that the Gamma function has applications in differential equations, but it is applied as such in, among others, the Airy function, Bessel functions ( see also Bessel-Clifford function ), the hypergeometric differential equation, Laguerre polynomials, and Meijer's Talk 06 : 49, 21 November 2006 ( UTC)
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