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Complex color plot of the Laguerre polynomial L n(x) with n as -1 divided by 9 and x as z to the power of 4 from -2-2i to 2+2i. In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of Laguerre's differential equation: ″ + ′ + =, = which is a second-order linear differential equation.
More generally, one can also consider integrands that have a known power-law singularity at x=0, for some real number >, leading to integrals of the form: + (). In this case, the weights are given [2] in terms of the generalized Laguerre polynomials:
In numerical analysis, Laguerre's method is a root-finding algorithm tailored to polynomials. In other words, Laguerre's method can be used to numerically solve the equation p ( x ) = 0 for a given polynomial p ( x ) .
The most general Laguerre-like polynomials, after the domain has been shifted and scaled, are the Associated Laguerre polynomials (also called generalized Laguerre polynomials), denoted (). There is a parameter , which can be any real number strictly greater than −1. The parameter is put in parentheses to avoid confusion with an exponent.
In mathematics, the q-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials P (α) n (x;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by Daniel S. Moak . Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties.
Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc – 14 August 1886, Bar-le-Duc) was a French mathematician [1] and a member of the Académie des sciences (1885). His main works were in the areas of geometry and complex analysis. He also investigated orthogonal polynomials (see Laguerre polynomials).
In mathematics, the little q-Laguerre polynomials p n (x;a|q) or Wall polynomials W n (x; b,q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme closely related to a continued fraction studied by Wall . (The term "Wall polynomial" is also used for an unrelated Wall polynomial in
If all roots of a polynomial are real, Laguerre proved the following lower and upper bounds of the roots, by using what is now called Samuelson's inequality. [ 17 ] Let ∑ k = 0 n a k x k {\displaystyle \sum _{k=0}^{n}a_{k}x^{k}} be a polynomial with all real roots.