Touchard polynomials

Summary

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.[1][2][3][4]

Properties

Basic 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:

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.

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:

Zeroes

All zeroes of the Touchard polynomials are real and negative. This fact was observed by L. H. Harper in 1967.[5]

The smallest zero is bounded from below (in absolute value) by[6]

although it is conjectured that the smallest zero grows linearly with the index n.

The Mahler measure of the Touchard polynomials can be estimated as follows:[7]

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

  1. ^ Roman, Steven (1984). The Umbral Calculus. Dover. ISBN 0-486-44139-3.
  2. ^ Boyadzhiev, Khristo N. "Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals". Abstract and Applied Analysis. 2009: 1–18. arXiv:0909.0979. Bibcode:2009AbApA2009....1B. doi:10.1155/2009/168672.
  3. ^ Brendt, Bruce C. "RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU" (PDF). Retrieved 23 November 2013.
  4. ^ Weisstein, Eric W. "Bell Polynomial". MathWorld.
  5. ^ Harper, L. H. (1967). "Stirling behavior is asymptotically normal". The Annals of Mathematical Statistics. 38 (2): 410–414. doi:10.1214/aoms/1177698956.
  6. ^ Mező, István; Corcino, Roberto B. (2015). "The estimation of the zeros of the Bell and r-Bell polynomials". Applied Mathematics and Computation. 250: 727–732. doi:10.1016/j.amc.2014.10.058.
  7. ^ István, Mező. "On the Mahler measure of the Bell polynomials". Retrieved 7 November 2017.
  • Touchard, Jacques (1939), "Sur les cycles des substitutions", Acta Mathematica, 70 (1): 243–297, doi:10.1007/BF02547349, ISSN 0001-5962, MR 1555449