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{{for|a different family of polynomials Q<sub>n</sub> occasionally called Touchard polynomials|Bateman polynomials}}
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The '''Touchard polynomials''', studied by {{harvs|txt|authorlink=Jacques Touchard|first=Jacques|last= Touchard|year=1939}}, also called the '''exponential polynomials''' in <ref name=Roman>{{cite book|last=Roman|first=Steven|title=The Umbral Calculus|year=1984|publisher=Dover|isbn=0-486-44139-3}}</ref><ref>{{cite web|last=Boyadzhiev|first=Khristo N.|title=Exponential polynomials, Stirling numbers, and evaluation of some gamma integrals.|url=http://arxiv.org/pdf/0909.0979.pdf|publisher=arxiv|accessdate=23 November 2013}}</ref>
,<ref>{{cite web|last=Brendt|first=Bruce C|title=RAMANUJAN REACHES HIS HAND FROM HIS GRAVE TO SNATCH YOUR THEOREMS FROM YOU|url=http://www.math.uiuc.edu/~berndt/articles/gravesnatching.pdf|accessdate=23 November 2013}}</ref> comprise a [[polynomial sequence]] of [[binomial type]] defined by
 
:<math>T_0(x) = 1,\qquad T_n(x)=\sum_{k=1}^n S(n,k)x^k=\sum_{k=1}^n
\left\{\begin{matrix} n \\ k \end{matrix}\right\}x^k, \quad n > 0,</math>
 
where ''S''(''n'', ''k'') is a [[Stirling number of the second kind]], i.e., it is the number of [[partition of a set|partitions of a set]] of size ''n'' into ''k'' disjoint non-empty subsets. (The second notation above, with { braces }, was introduced by [[Donald Knuth]].)  The value at 1 of the ''n''th Touchard polynomial is the ''n''th [[Bell numbers|Bell number]], i.e., the number of [[partition of a set|partitions of a set]] of size ''n'':
 
:<math>T_n(1)=B_n.</math>
 
If ''X'' is a [[random variable]] with a [[Poisson distribution]] with expected value λ, then its ''n''th moment is E(''X''<sup>''n''</sup>) = ''T''<sub>''n''</sub>(λ), leading to the definition:
 
:<math>T_{n}(x)=e^{-x}\sum_{k=0}^\infty \frac {x^k k^n} {k!}.</math>
 
Using this fact one can quickly prove that this [[polynomial sequence]] is of [[binomial type]], i.e., it satisfies the sequence of identities:
 
:<math>T_n(\lambda+\mu)=\sum_{k=0}^n {n \choose k} T_k(\lambda) T_{n-k}(\mu).</math>
 
The Touchard polynomials make up the only polynomial sequence of binomial type in which the coefficient of the 1st-degree term of every polynomial is 1.
 
:<math>T_{n+1}(x)=x\sum_{k=0}^n{n \choose k}T_k(x).</math>
 
The Touchard polynomials satisfy the Rodrigues-like formula:
 
:<math>T_n \left(e^x \right) = e^{-e^x} \frac{d^n}{dx^n}\left(e^{e^x}\right)</math>
 
The Touchard polynomials satisfy the [[recurrence relation]]
 
:<math>T_{n+1}(x)=x \left(1+\frac{d}{dx} \right)T_{n}(x).</math>
 
And
 
:<math>T_{n+1}(x)=x\sum_{k=0}^n{n \choose k}T_k(x).</math>
 
In case ''x'' = 1, this reduces to the recurrence formula for the [[Bell numbers]].
 
Using the Umbral notation ''T''<sup>''n''</sup>(''x'')=''T''<sub>''n''</sub>(''x''),these formulas become:
 
:<math>T_n(\lambda+\mu)=\left(T(\lambda)+T(\mu) \right)^n .</math>
 
:<math>T_{n+1}(x)=x \left(1+T(x) \right)^n.</math>
 
The [[generating function]] of the Touchard polynomials is
 
:<math>\sum_{n=0}^\infty {T_n(x) \over n!} t^n=e^{x\left(e^t-1\right)}.</math>
 
This corresponds to the generating function of [[Stirling numbers of the second kind#Generating function]] and <ref name=Roman>{{cite book|last=Roman|first=Steven|title=The Umbral Calculus|year=1984|publisher=Dover|isbn=0-486-44139-3|pages=63–64}}</ref> where it is referred to as Exponential Polynomials.
And a contour-integral representation is
 
:<math>T_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{x({e^t}-1)}}{t^{n+1}}\,dt</math>
 
The Touchard polynomials (and thereby the [[Bell numbers]]) can be generalized, using the real part of the above integral, to non-integer order:
 
:<math>T_n(x)=\frac{n!}{\pi} \int^{\pi}_0 e^{x \bigl(e^{\cos(\theta)} \cos(\sin(\theta))-1 \bigr)} \cos \bigl(x e^{\cos(\theta)} \sin(\sin(\theta)) -n\theta) \, \mathrm{d}\theta </math>
 
==References==
{{Reflist}}
 
*{{Citation | last1=Touchard | first1=Jacques | title=Sur les cycles des substitutions | doi=10.1007/BF02547349 | mr=1555449 | year=1939 | journal=[[Acta Mathematica]] | issn=0001-5962 | volume=70 | issue=1 | pages=243–297}}
 
{{DEFAULTSORT:Touchard Polynomials}}
[[Category:Polynomials]]

Latest revision as of 16:34, 29 December 2014

If you have the desire to process settings immediately, loading files quickly, however the body is logy plus torpid, what would you do? If you are a giant "switchboard" which is deficiency of effective management system and powerful housekeeper, what would we do? If you have send your exact commands to your notice, nevertheless the body could not perform correctly, what would you do? Yes! You want a full-featured repair registry!

Google Chrome crashes on Windows 7 by the corrupted cache contents and issues with all the stored browsing data. Delete the browsing data and clear the contents of the cache to resolve this problem.

So, this advanced double scan is not only 1 of the better, yet it is equally freeware. And as of all of this which various regard CCleaner among the better registry products inside the marketplace today. I would add which I personally choose Regcure for the easy reason which it has a better interface plus I understand for a fact which it is ad-ware without charge.

Paid registry products found on the other hand, I have found, are often cheap. They supply normal, free changes or at least cheap changes. This follows considering the software producer requires to guarantee their product is best inside staying ahead of its competitors.

Use a tuneup utilities 2014. This can look the Windows registry for 3 kinds of keys which can definitely hurt PC performance. These are: duplicate, missing, plus corrupted.

Let's start with all the bad sides first. The initial cost of the product is especially cheap. However, it only comes with 1 year of updates. After that you must register to monthly updates. The advantage of that is the fact that best optimizer has enough funds and resources to research mistakes. This means, you may be ensured of secure fixes.

Your disk demands area in purchase to run smoothly. By freeing up certain area from the disk, you are able to speed up the PC a bit. Delete all file in the temporary web files folder, recycle bin, well-defined shortcuts and icons from a desktop that you never utilize plus remove programs we do not use.

By changing the means you utilize the web you are able to have access more of the valuable bandwidth. This can eventually give you a faster surfing experience. Here is a link to 3 ways to personalize the PC speed online.

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