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gptkbp:instanceOf
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gptkb:orthogonal_polynomials
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gptkbp:category
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special functions
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gptkbp:degree
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n
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gptkbp:differential
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difference equation, not differential equation
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gptkbp:domain
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non-negative integers
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gptkbp:field
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gptkb:mathematics
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gptkbp:generatingFunction
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e^{-a t}(1+t)^x = \sum_{n=0}^\infty C_n(x;a) \frac{t^n}{n!}
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gptkbp:hasSpecialCase
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gptkb:Meixner_polynomials
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gptkbp:hypergeometricRepresentation
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C_n(x;a) = {}_2F_0(-n,-x;-;-1/a)
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gptkbp:introducedIn
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1905
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gptkbp:namedAfter
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gptkb:Carl_Charlier
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gptkbp:orthogonalityRelation
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\sum_{x=0}^\infty C_m(x;a)C_n(x;a)\frac{a^x}{x!}e^{-a} = a^n n! \delta_{mn}
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gptkbp:orthogonalWithRespectTo
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gptkb:Poisson_distribution
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gptkbp:parameter
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a
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gptkbp:recurrence
|
C_{n+1}(x;a) = (x-n)C_n(x;a) - aC_n(x-1;a)
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gptkbp:relatedTo
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gptkb:Laguerre_polynomials
gptkb:Meixner_polynomials
Krawtchouk polynomials
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gptkbp:type
|
discrete orthogonal polynomials
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gptkbp:usedIn
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gptkb:combinatorics
gptkb:probability_theory
statistics
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gptkbp:variant
|
x
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gptkbp:weight
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w(x) = \frac{a^x}{x!}e^{-a}
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gptkbp:bfsParent
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gptkb:Carl_Charlier
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gptkbp:bfsLayer
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7
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https://www.w3.org/2000/01/rdf-schema#label
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Charlier polynomials
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