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