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Touchard polynomials

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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

Contents

T n ( x ) = ∑ k = 0 n S ( n , k ) x k = ∑ k = 0 n { n k } x k ,

where S ( n , k ) = { n k } 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.

Properties

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:

T n ( 1 ) = B 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:

T n ( x ) = e − x ∑ k = 0 ∞ x k k n k ! .

Using this fact one can quickly prove that this polynomial sequence is of binomial type, i.e., it satisfies the sequence of identities:

T n ( λ + μ ) = ∑ k = 0 n ( n k ) T k ( λ ) T n − k ( μ ) .

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:

T n ( e x ) = e − e x d n d x n ( e e x )

The Touchard polynomials satisfy the recurrence relation

T n + 1 ( x ) = x ( 1 + d d x ) T n ( x )

and

T n + 1 ( x ) = x ∑ k = 0 n ( n k ) T k ( x ) .

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:

T n ( λ + μ ) = ( T ( λ ) + T ( μ ) ) n , T n + 1 ( x ) = x ( 1 + T ( x ) ) n .

The generating function of the Touchard polynomials is

∑ n = 0 ∞ T n ( x ) n ! t n = e x ( e t − 1 ) ,

which corresponds to the generating function of Stirling numbers of the second kind.

Touchard polynomials have contour integral representation:

T n ( x ) = n ! 2 π i ∮ e x ( e t − 1 ) t n + 1 d t .

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

1 n ( n 2 ) + n − 1 n ( n 2 ) 2 − 2 n n − 1 ( ( n 3 ) + 3 ( n 4 ) ) , although it is believed by the same authors that the leftmost root grows linearly with the index n.

Generalizations

  • Complete Bell polynomial B n ( x 1 , x 2 , … , x n ) may be viewed as a multivariate generalization of Touchard polynomial T n ( x ) , since T n ( x ) = B n ( x , x , … , x ) .
  • The Touchard polynomials (and thereby the Bell numbers) can be generalized, using the real part of the above integral, to non-integer order:
    • T n ( x ) = n ! π ∫ 0 π e x ( e cos ⁡ ( θ ) cos ⁡ ( sin ⁡ ( θ ) ) − 1 ) cos ⁡ ( x e cos ⁡ ( θ ) sin ⁡ ( sin ⁡ ( θ ) ) − n θ ) d θ

    References

    Touchard polynomials Wikipedia
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