Millman's theorem: Difference between revisions
en>Pinethicket m Reverted edits by 210.212.53.139 (talk) to last version by Luckas-bot |
en>Minketan Patel No edit summary |
||
| Line 1: | Line 1: | ||
{{Noref|date=August 2011}} | |||
'''Jacques Touchard''' (1885 – 1968) was a [[French people|French]] [[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]]. He is also known for his solution to the [[ménage problem]] of counting seating arrangements in which men and women alternate and are not seated next to their spouses. | |||
==Touchard's Catalan identity== | |||
The following algebraic identity involving the [[Catalan number]]s | |||
:<math> C_k ={ 1\over{k+1}}{{2k}\choose {k}},\quad k \ge 0 </math> | |||
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 plenitude of different definitions for the Catalan numbers). | |||
For ''n'' ≥ 0 one has | |||
:<math> C_{n+1} =\sum_{k \,\le\, n/2} 2^{n-2k} {n \choose 2k} C_k. \,</math> | |||
Using the [[generating function]] | |||
:<math> C(t)=\sum_{n \ge 0} C_n t^n ={{1-\sqrt{1-4t}}\over {2t}} </math> | |||
it can be proved by algebraic manipulations of generating [[series (mathematics)|series]] that Touchard's identity is | |||
equivalent to the [[functional equation]] | |||
:<math> {t \over {1-2t}} C\left({t^2\over (1-2t)^2}\right) = C(t)-1 </math> | |||
satisfied by the Catalan generating series ''C''(''t''). | |||
{{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. --> | |||
| NAME = Touchard, Jacques | |||
| ALTERNATIVE NAMES = | |||
| SHORT DESCRIPTION = French mathematician | |||
| DATE OF BIRTH = 1885 | |||
| PLACE OF BIRTH = | |||
| DATE OF DEATH = 1968 | |||
| PLACE OF DEATH = | |||
}} | |||
{{DEFAULTSORT:Touchard, Jacques}} | |||
[[Category:French mathematicians]] | |||
[[Category:1885 births]] | |||
[[Category:1968 deaths]] | |||
[[Category:Place of birth missing]] | |||
{{france-mathematician-stub}} | |||
Revision as of 07:46, 20 December 2013
Template:Noref Jacques Touchard (1885 – 1968) was a French 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. He is also known for his solution to the ménage problem of counting seating arrangements in which men and women alternate and are not seated next to their spouses.
Touchard's Catalan identity
The following algebraic identity involving the Catalan numbers
is apparently due to Touchard (according to Richard P. Stanley, who mentions it in his panorama article "Exercises on Catalan and Related Numbers" giving an overwhelming plenitude of different definitions for the Catalan numbers). For n ≥ 0 one has
Using the generating function
it can be proved by algebraic manipulations of generating series that Touchard's identity is equivalent to the functional equation
satisfied by the Catalan generating series C(t).