Touchard polynomials

Sequence of polynomials

For a different family of polynomials Qn occasionally called Touchard polynomials, see Bateman polynomials.

Not to be confused with Bell polynomials.

Touchard Polynomials

The Touchard polynomials, studied by Jacques Touchard (1939),[1] also called the exponential polynomials or Bell polynomials, comprise a polynomial sequence of binomial type defined by

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

where  S ( n , k ) = { n k } {\displaystyle 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.[2][3][4][5]

The first few Touchard polynomials are

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

## Properties

[edit]

### Basic properties

[edit]

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 . {\displaystyle 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 ! . {\displaystyle T_{n}(x)=e^{-x}\sum _{k=0}^{\infty }{\frac {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 ) . {\displaystyle T_{n}(\lambda +\mu )=\sum _{k=0}^{n}{n \choose k}T_{k}(\lambda )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 ( e x ) = e − e x d n d x n e e x . {\displaystyle T_{n}\left(e^{x}\right)=e^{-e^{x}}{\frac {d^{n}}{dx^{n}}}\;e^{e^{x}}.}

The Touchard polynomials satisfy the recurrence relation

T n + 1 ( x ) = x ( 1 + d d x ) T n ( x ) {\displaystyle T_{n+1}(x)=x\left(1+{\frac {d}{dx}}\right)T_{n}(x)}

and

T n + 1 ( x ) = x ∑ k = 0 n ( n k ) T k ( x ) . {\displaystyle T_{n+1}(x)=x\sum _{k=0}^{n}{n \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[6]

T n + m ( x ) = ∑ k = 0 n { n k } x k ∑ j = 0 m ( m j ) k m − j T j ( x ) {\displaystyle T_{n+m}(x)=\sum _{k=0}^{n}\left\\{{n \atop k}\right\\}x^{k}\sum _{j=0}^{m}{\binom {m}{j}}k^{m-j}T_{j}(x)}

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

T n ( λ + μ ) = ( T ( λ ) + T ( μ ) ) n , {\displaystyle T_{n}(\lambda +\mu )=\left(T(\lambda )+T(\mu )\right)^{n},} [clarification needed]
T n + 1 ( x ) = x ( 1 + T ( x ) ) n . {\displaystyle T_{n+1}(x)=x\left(1+T(x)\right)^{n}.}

The generating function of the Touchard polynomials is

∑ n = 0 ∞ T n ( x ) n ! t n = e x ( e t − 1 ) , {\displaystyle \sum _{n=0}^{\infty }{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 π i ∮ e x ( e t − 1 ) t n + 1 d t . {\displaystyle T_{n}(x)={\frac {n!}{2\pi i}}\oint {\frac {e^{x({e^{t}}-1)}}{t^{n+1}}}\,dt.}

### Zeroes

[edit]

All zeroes of the Touchard polynomials are real and negative. This fact was observed by L. H. Harper in 1967.[7]

The absolute value of the leftmost zero is bounded from above by[8]

1 n ( n 2 ) + n − 1 n ( n 2 ) 2 − 2 n n − 1 ( ( n 3 ) + 3 ( n 4 ) ) , {\displaystyle {\frac {1}{n}}{\binom {n}{2}}+{\frac {n-1}{n}}{\sqrt {{\binom {n}{2}}^{2}-{\frac {2n}{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.

The Mahler measure M ( T n ) {\displaystyle M(T_{n})} of the Touchard polynomials can be estimated as follows:[9]

{ n Ω n } ( n Ω n ) ≤ M ( T n ) ≤ n + 1 { n K n } , {\displaystyle {\frac {\lbrace \textstyle {n \atop \Omega _{n}}\rbrace }{\binom {n}{\Omega _{n}}}}\leq M(T_{n})\leq {\sqrt {n+1}}\left\\{{n \atop K_{n}}\right\\},}

where  Ω n {\displaystyle \Omega _{n}} and  K n {\displaystyle K_{n}} are the smallest of the maximum two k indices such that  { n k } / ( n k ) {\displaystyle \lbrace \textstyle {n \atop k}\rbrace /{\binom {n}{k}}} and  { n k } {\displaystyle \lbrace \textstyle {n \atop k}\rbrace } are maximal, respectively.

## Generalizations

[edit]

* Complete Bell polynomial B n ( x 1 , x 2 , … , x n ) {\displaystyle B_{n}(x_{1},x_{2},\dots ,x_{n})} may be viewed as a multivariate generalization of Touchard polynomial  T n ( x ) {\displaystyle T_{n}(x)} , since  T n ( x ) = B n ( x , x , … , x ) . {\displaystyle T_{n}(x)=B_{n}(x,x,\dots ,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 . {\displaystyle T_{n}(x)={\frac {n!}{\pi }}\int _{0}^{\pi }e^{x{\bigl (}e^{\cos(\theta )}\cos(\sin(\theta ))-1{\bigr )}}\cos {\bigl (}xe^{\cos(\theta )}\sin(\sin(\theta ))-n\theta {\bigr )}\,d\theta \,.}

## See also

[edit]

* Bell polynomials

## References

[edit]

1. ^ Touchard, Jacques (1939), "Sur les cycles des substitutions", Acta Mathematica, 70 (1): 243–297, doi:10.1007/BF02547349, ISSN 0001-5962, MR 1555449
2. ^ Roman, Steven (1984). The Umbral Calculus. Dover. ISBN 0-486-44139-3.
3. ^ Boyadzhiev, Khristo N. (2009). "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.
4. ^ Brendt, Bruce C. "RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU" (PDF). Retrieved 23 November 2013.
5. ^ Weisstein, Eric W. "Bell Polynomial". MathWorld.
6. ^ "Implications of Spivey's Bell Number Formula". cs.uwaterloo.ca. Retrieved 2023-05-28.
7. ^ Harper, L. H. (1967). "Stirling behavior is asymptotically normal". The Annals of Mathematical Statistics. 38 (2): 410–414\. doi:10.1214/aoms/1177698956.
8. ^ 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.
9. ^ István, Mező. "On the Mahler measure of the Bell polynomials". Retrieved 7 November 2017.