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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.
When a is a non-positive integer, then Kummer's function (if it is defined) is a generalized Laguerre polynomial. Just as the confluent differential equation is a limit of the hypergeometric differential equation as the singular point at 1 is moved towards the singular point at ∞, the confluent hypergeometric function can be given as a limit ...
The polynomials can be factored into linear factors of the form ... An example is the power series for the exponential function. ... these are the Laguerre polynomials.
Exponential generating function (,)! =! ... This formula is the default Laguerre polynomial in Umbral calculus convention. [8] Practical application ...
In numerical analysis Gauss–Laguerre quadrature (named after Carl Friedrich Gauss and Edmond Laguerre) is an extension of the Gaussian quadrature method for approximating the value of integrals of the following kind: + (). In this case
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.
Interesting variants can be obtained by obvious modification of the matrix-logarithm PL 7 and then application of the matrix exponential. The first example below uses the squares of the values of the log-matrix and constructs a 7 × 7 "Laguerre"- matrix (or matrix of coefficients of Laguerre polynomials
The exponential generating function of a sequence a n is ... the Bell numbers, B(n), the Laguerre polynomials, and the Stirling convolution polynomials. ...