Jacques Touchard

Jacques Touchard (1885 – 1968) is a mathematician. In 1953, he proved that an odd perfect number must be of the form 12"k" + 1 or 36"k" + 9. In combinatorics and probability theory, he introduced the Touchard polynomials.

Touchard's Catalan identity

The following algebraic identity involving the Catalan numbers

:$C\_k\; =\{\; 1over\{k+1$2k}choose {k, , k ge 0

is apparently due to Touchard (according to Richard P. Stanley, who mentions it in his panorama article " [http://www-math.mit.edu/~rstan/ec/catalan.pdf Exercises on Catalan and Related Numbers] " giving an overwhelming plentitude of different definitions for the Catalan numbers).For $n\; ge\; 0$ one has :$C\_\{n+1\}\; =sum\_\{k\; le\; \{nover\; 2\; 2^\{n-2k\}\; \{n\; choose\; 2k\}\; C\_\{k\}\; .$Using the generating function :$C(t)=sum\_\{n\; ge\; 0\}\; C\_n\; t^n\; =$1-sqrt{1-4tover {2t it can be proved by algebraic manipulations of generating series that Touchard's identity is equivalent to the functional equation:$$t}over{1-2tCleft(t^2}over{(1-2t)^2 ight)=C(t)-1 satisfied by the Catalan generating series "C"("t").