Laguerre Polynomials

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Laguerre polynomials

[lə′ger ‚päl·ə′nō·mē·əlz]
A sequence of orthogonal polynomials which solve Laguerre's differential equation for positive integral values of the parameter.

Laguerre Polynomials


(named after the French mathematician E. Laguerre, 1834–86), a special system of polynomials of successively increasing degree. For n = 0, 1, 2, … the Laguerre polynomials Ln(x) can be defined by the formula

The first few Laguerre polynomials are

L0(x) = 1, L1(x) = x – 1, l2(x) = x2 – 4x + 2

L3(x) = x3 – 9x2 + 18x – 6

The Laguerre polynomials are orthogonal on the half-line x ≥ 0 with respect to the weight function e–x and are solutions of the differential equation

xyn + (1 – x)y + ny = 0

The Laguerre polynomials are recursively defined by the formula

Ln+1(x) = (x – 2n – 1)Ln(x) – n2Ln–1(x)


Lebedev, N. N. Spetsial’nye funktsii i ikh prilozheniia, 2nd ed. Moscow-Leningrad, 1963.
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Later, Laguerre function approximation is also introduced into the time domain reduction and it has been proven that in this case there exists an equivalent relationship to moment matching methods in the frequency domain [14-17].
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The aim of this paper is to develop the mixed generalized Laguerre-Legendre pseudospectral method for non-isotropic heat transfer in an infinite strip, by using the Legendre interpolation in the direction of finite length, and the generalized Laguerre function interpolation in the infinite long direction.
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Wang [26] studied the Fisher equation on a semi-infinite domain using the generalized Laguerre functions. Wang and Jiao [27] considered the Fisher equation on unbounded domain using the generalized Hermite functions.
In particular, the polyBergman spaces are spaces of Wavelet transform which is related to Laguerre functions, and the poly-Fock spaces are spaces of short-time Fourier transform which is related to Hermite functions.