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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 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.
The exponential generating function of a sequence a n is ... the Bell numbers, B(n), the Laguerre polynomials, and the Stirling convolution polynomials. ...
Laguerre's method may even converge to a complex root of the polynomial, because the radicand of the square root may be of a negative number, in the formula for the correction, , given above – manageable so long as complex numbers can be conveniently accommodated for the calculation. This may be considered an advantage or a liability ...
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