Sequence of polynomials
For a different family of polynomials Q
n 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
The first few Touchard polynomials are
Properties
Basic properties
The value at 1 of the nth Touchard polynomial is the nth
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
and
In the case x = 1, this reduces to the recurrence formula for the
Bell numbers
.
A generalization of both this formula and the definition, is a generalization of Spivey's formula[5]
Using the umbral notation Tn(x)=Tn(x), these formulas become:
- [clarification needed]
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:
Zeroes
All zeroes of the Touchard polynomials are real and negative. This fact was observed by L. H. Harper in 1967.[6]
The absolute value of the leftmost zero is bounded from above by[7]
although it is conjectured that the leftmost zero grows linearly with the index n.
The Mahler measure of the Touchard polynomials can be estimated as follows:[8]
where and are the smallest of the maximum two k indices such that
and
are maximal, respectively.
Generalizations
- 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:
See also
References
- Touchard, Jacques (1939), "Sur les cycles des substitutions",