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gptkbp:instanceOf
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orthogonal polynomials
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gptkbp:application
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gptkb:optics
gptkb:probability_theory
gptkb:signal_processing
gptkb:Laguerre–Gaussian_modes
statistics
combinatorics
random matrix theory
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gptkbp:author
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gptkb:Edmond_Laguerre
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gptkbp:category
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special functions
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gptkbp:domain
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complex numbers
real numbers
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gptkbp:field
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gptkb:mathematics
mathematical physics
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gptkbp:firstAppearance
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1880s
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gptkbp:firstPolynomial
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L_0(x) = 1
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gptkbp:hasGeneratingFunction
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(1-t)^{-1} exp(-xt/(1-t))
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gptkbp:hasRodriguesFormula
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L_n(x) = exp(x)/n! d^n/dx^n (exp(-x)x^n)
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gptkbp:hasSpecialCase
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gptkb:generalized_Laguerre_polynomials
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gptkbp:hasWikipediaPage
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https://en.wikipedia.org/wiki/Laguerre_polynomials
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gptkbp:hasZeros
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all real and positive
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https://www.w3.org/2000/01/rdf-schema#label
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Laguerre polynomials
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gptkbp:namedAfter
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gptkb:Edmond_Laguerre
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gptkbp:notation
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L_n(x)
L_n^{(α)}(x) for generalized case
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gptkbp:orthogonalOn
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[0, ∞)
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gptkbp:orthogonalWithRespectTo
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weight function exp(-x) on [0, ∞)
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gptkbp:recurrence
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(n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x)
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gptkbp:relatedTo
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gptkb:Jacobi_polynomials
gptkb:Charlier_polynomials
gptkb:Hermite_polynomials
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gptkbp:satisfies
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gptkb:Laguerre_differential_equation
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gptkbp:secondPolynomial
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L_1(x) = 1 - x
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gptkbp:sequence
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L_0(x), L_1(x), L_2(x), ...
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gptkbp:thirdPolynomial
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L_2(x) = 1 - 2x + x^2/2
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gptkbp:usedIn
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numerical analysis
quantum mechanics
approximation theory
solution of the radial part of the hydrogen atom
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gptkbp:weight
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exp(-x)
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gptkbp:bfsParent
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gptkb:Hermite_polynomials
gptkb:Generalized_Polynomial_Chaos
gptkb:Sturm–Liouville_problem
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gptkbp:bfsLayer
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6
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