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- Touchard_polynomials abstract "The Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials in , comprise a polynomial sequence of binomial type defined bywhere S(n, k) is a Stirling number of the second kind, i.e., it is the number of partitions of a set of size n into k disjoint non-empty subsets. (The second notation above, with { braces }, was introduced by Donald Knuth.) The value at 1 of the nth Touchard polynomial is the nth Bell number, i.e., the number of partitions of a set of size n:If X is a random variable with a Poisson distribution with expected value λ, then its nth moment is E(Xn) = Tn(λ), leading to the definition:Using this fact one can quickly prove that this polynomial sequence is of binomial type, i.e., it satisfies the sequence of identities:The Touchard polynomials make up the only polynomial sequence of binomial type in which the coefficient of the 1st-degree term of every polynomial is 1.The Touchard polynomials satisfy the Rodrigues-like formula:The Touchard polynomials satisfy the recurrence relationAndIn case x = 1, this reduces to the recurrence formula for the Bell numbers.Using the Umbral notation Tn(x)=Tn(x),these formulas become:The generating function of the Touchard polynomials isThis corresponds to the generating function of Stirling numbers of the second kind#Generating function and where it is referred to as Exponential Polynomials.And a contour-integral representation isThe Touchard polynomials (and thereby the Bell numbers) can be generalized, using the real part of the above integral, to non-integer order:".
- Touchard_polynomials wikiPageID "342592".
- Touchard_polynomials wikiPageRevisionID "583068649".
- Touchard_polynomials authorlink "Jacques Touchard".
- Touchard_polynomials first "Jacques".
- Touchard_polynomials hasPhotoCollection Touchard_polynomials.
- Touchard_polynomials last "Touchard".
- Touchard_polynomials year "1939".
- Touchard_polynomials subject Category:Polynomials.
- Touchard_polynomials type Abstraction100002137.
- Touchard_polynomials type Function113783816.
- Touchard_polynomials type MathematicalRelation113783581.
- Touchard_polynomials type Polynomial105861855.
- Touchard_polynomials type Polynomials.
- Touchard_polynomials type Relation100031921.
- Touchard_polynomials comment "The Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials in , comprise a polynomial sequence of binomial type defined bywhere S(n, k) is a Stirling number of the second kind, i.e., it is the number of partitions of a set of size n into k disjoint non-empty subsets.".
- Touchard_polynomials label "Polinomios de Touchard".
- Touchard_polynomials label "Touchard polynomials".
- Touchard_polynomials sameAs Polinomios_de_Touchard.
- Touchard_polynomials sameAs トゥシャール多項式.
- Touchard_polynomials sameAs m.01y92j.
- Touchard_polynomials sameAs Q3820608.
- Touchard_polynomials sameAs Q3820608.
- Touchard_polynomials sameAs Touchard_polynomials.
- Touchard_polynomials wasDerivedFrom Touchard_polynomials?oldid=583068649.
- Touchard_polynomials isPrimaryTopicOf Touchard_polynomials.