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  2. Laguerre polynomials - Wikipedia

    en.wikipedia.org/wiki/Laguerre_polynomials

    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.

  3. Gauss–Laguerre quadrature - Wikipedia

    en.wikipedia.org/wiki/Gauss–Laguerre_quadrature

    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:

  4. Laguerre's method - Wikipedia

    en.wikipedia.org/wiki/Laguerre's_method

    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 ) .

  5. Classical orthogonal polynomials - Wikipedia

    en.wikipedia.org/wiki/Classical_orthogonal...

    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.

  6. q-Laguerre polynomials - Wikipedia

    en.wikipedia.org/wiki/Q-Laguerre_polynomials

    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.

  7. Edmond Laguerre - Wikipedia

    en.wikipedia.org/wiki/Edmond_Laguerre

    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).

  8. Little q-Laguerre polynomials - Wikipedia

    en.wikipedia.org/wiki/Little_q-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

  9. Geometrical properties of polynomial roots - Wikipedia

    en.wikipedia.org/wiki/Geometrical_properties_of...

    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.