Jacques Touchard

From Wikipedia, the free encyclopedia

Jacques Touchard is a mathematician. In 1953, he proved that an odd perfect number must be of the form 12k + 1 or 36k + 9. In combinatorics and probability theory, he introduced the Touchard polynomials.

[edit] Touchard's Catalan identity

The following algebraic identity involving the Catalan numbers

C_k ={ 1\over{k+1}}{{2k}\choose {k}}\, , k \ge 0

is apparently due to Jacques Touchard (according to Richard P. Stanley, who mentions it in his panorama article "Exercises on Catalan and Related Numbers" giving an overwhelming plentitude of different definitions for the Catalan numbers). For n \ge 0 one has

C_{n+1} =\sum_{k \le {n\over 2}} 2^{n-2k} {n \choose 2k} C_{k} \ .

Using the generating function

C(t)=\sum_{n \ge 0} C_n t^n ={{1-\sqrt{1-4t}}\over {2t}}

it can be proved by algebraic manipulations of generating series that Touchard's identity is equivalent to the functional equation

{{t}\over{1-2t}}C\left({{t^2}\over{(1-2t)^2}}\right)=C(t)-1

satisfied by the Catalan generating series C(t).


This article about a mathematician is a stub. You can help Wikipedia by expanding it.
Retrieved from "http://en.wikipedia.org../../../j/a/c/Jacques_Touchard_a5ea.html"
In other languages