Touchard polynomials explained

Touchard polynomials should not be confused with Bell polynomials.

The Touchard polynomials, studied by, also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by

T _{n(x)=\sum} ^{n} ^{k =0} S(n,k)x ^{n \left\{} _{k =0} {n \atop k} \right\}x ^{k,}

where

S(n,k) = \left\{ {n \atop k} \right\} 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]}

The first few Touchard polynomials are

T _{1(x)=x,} T ^{2+ x,} _{2 (x) = x} T ^{3+3 x} _{3 (x) = x} ^{2+x,} T ^{4+6 x} _{4 (x) = x} ^{3+7x} ^{2+x,} T ^{5+10 x} _{5 (x) = x} ^{4+25x} ^{3+15x} ^{2+x.}

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:

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) = T_n(λ), leading to the definition:

T _{n} (x) = e ^{-x} \sum ^{inf ty} _{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+\mu)=\sum} ^{n} ^{k =0} {n \choose k} T _{k(λ)} T _{n-k} (\mu).

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} \left(e ^{x} \right) = e ^{- e ^{x}} (d ^{n})/(dx ^{n}) e ^{e ^{x}} .

The Touchard polynomials satisfy the recurrence relation

T _{n+1} (x) = x \left( 1+ (d)/(dx) \right)T _{n} (x) andT _{n+1} (x) = x\sum ^{n{n} _{k =0} \choose k}T _{k(x).} 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^[4]

$$T\_(x)\; =\; \backslash sum\_^n\; \backslash left\backslash \; x^k\; \backslash sum\_^m\; \backslash binom\; k^\; T\_j(x)$$

Using the umbral notation Tⁿ(x)=T_n(x), these formulas become:

T _{n(λ+\mu)=\left(T(λ)+T(\mu)} \right) ^{n,} T _{n+1} (x) = x \left( 1+ T(x) \right) ^{n.}

The generating function of the Touchard polynomials is

\sum ^{inf ty} _{n =0} {T _{n(x)} \over n!} t ^{n=e} ^{x\left(e ^{t-1\right)}} , which corresponds to the generating function of Stirling numbers of the second kind.

Touchard polynomials have contour integral representation:

T _{n(x) = (n!)/(2 \pi i)} \oint (e ^{x({e ^{t}} -1 ))/() }\,dt.

Zeroes

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

The absolute value of the leftmost zero is bounded from above by^[6]

(1 n\binom{n}{ 2 } + (n -1)/(n))/(\sqrt{\binom{n}{ 2 }) ^{2- (2 n)/(n -1)} \left(\binom{n}{ 3 } +3 \binom{n}{ 4 }\right)}, although it is conjectured that the leftmost zero grows linearly with the index n.M(T _{n)} of the Touchard polynomials can be estimated as follows:^[7] (\lbrace style{n\atop \Omega _{n})/(\rbrace}{\binom{n}{\Omega) _{n}}\le} M(T _{n)\le\sqrt{n+1}\left\{{n\atop} K _{n}\right\},} where \Omega _{n} and K _{n} are the smallest of the maximum two k indices such that\lbrace style{n\atop k}\rbrace/\binom{n}{k} and \lbrace style{n\atop k}\rbrace are maximal, respectively.

Generalizations

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!)/(\pi)} \int ^{\pi} _{0} e ^{x l (e ^{\cos(\theta)} \cos(\sin(\theta)) -1 r )} \cos l (x e ^{\cos(\theta)} \sin(\sin(\theta)) - n\theta r ) d\theta .

See also

• Bell polynomials

Notes and References

1. Book: Roman, Steven. The Umbral Calculus. 1984. Dover. 0-486-44139-3.
2. Boyadzhiev. Khristo N.. Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals . 0909.0979. 10.1155/2009/168672. 2009. Abstract and Applied Analysis. 2009 . 1–18. 2009AbApA2009....1B . free .
3. Web site: Brendt. Bruce C. RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU. 23 November 2013.
4. Web site: Implications of Spivey's Bell Number Formula . 2023-05-28 . cs.uwaterloo.ca.
5. Harper . L. H.. Stirling behavior is asymptotically normal. 1967. 38. 2. 410–414. The Annals of Mathematical Statistics . 10.1214/aoms/1177698956. free.
6. Mező . István. Corcino . Roberto B.. The estimation of the zeros of the Bell and r-Bell polynomials. 2015. 250. 727–732. Applied Mathematics and Computation . 10.1016/j.amc.2014.10.058.
7. Web site: István. Mező. On the Mahler measure of the Bell polynomials. 7 November 2017.