For a different family of polynomials Qn
occasionally called Touchard polynomials, see Bateman polynomials
The Touchard polynomials, studied by Jacques Touchard (1939), also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by
where is a Stirling number of the second kind, i.e., the number of partitions of a set of size n into k disjoint non-empty subsets.
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 constitute the only polynomial sequence of binomial type with the coefficient of x equal 1 in every polynomial.
The Touchard polynomials satisfy the Rodrigues-like formula:
The Touchard polynomials satisfy the recurrence relation
In the 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 is
which corresponds to the generating function of Stirling numbers of the second kind.
Touchard polynomials have contour integral representation:
The Touchard polynomials have only real and negative roots. This fact was proven by L. H. Harper in 1967. The leftmost root is bounded from below (in absolute value) by
although it is believed by the same authors that the leftmost root grows linearly with the index n.
- Complete Bell polynomial may be viewed as a multivariate generalization of Touchard polynomial , since
- The Touchard polynomials (and thereby the Bell numbers) can be generalized, using the real part of the above integral, to non-integer order: