Talk:Fuss–Catalan number

Latest comment: 10 years ago by Bbloodaxe in topic Suggestion 3

Suggestion1

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Couple of things to add that I'm not able to do properly... 1 - Am(1,x) this is Pascal's triangle (rotated), makes more sense when you plot it out. 2 - The generating function satisfies

 f(x) = (1+x.f(x)^(p/r))^r .. see Wojciech Mlotkowski's paper

This can give links to binomial distribution, and generating functions...

-- Above comments have been completed 13:45, 14 January 2014 (UTC) bbloodaxe  — Preceding unsigned comment added by Bbloodaxe (talkcontribs)  

Suggestion 2

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I'm interested in why the following was changed from:

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Let the ordinary generating function with respect to the index be defined as follows

 , then the Wojciech Mlotkowski paper (see references), shows that as
  then it directly follows that  .

This can extended by using Lambert's equivalence   to the general generating function, for all the Fuss-Catalan numbers:

 .

to

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Let the ordinary generating function with respect to the index be defined as follows

 .

An immediate consequence of the representation as a Gamma Function ratio is

 .

Then the Wojciech Mlotkowski paper (see references), shows that  . This can extended by using Lambert's equivalence   to:

 .

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I did not follow Mlotkowski exactly, instead I paraphrased it, and also checked with him that the generral generating function is correct. This was the case. That said I feel the current repreaentation needs improving.

  • 1 - the Mlotkowski paper makes no claim for Gamma Function ratio being involved with  , it comes from equations 2.2 & 2.3 in his paper. It may well be a consequence of such, but that was not the presented logic.
  • 2 - the Mlotkiski paper does not show  . This is a a logical consequence that  , and merely arrived at by substitution into the generating formula  . I will agree that the paper shows a similar formula, namely   equation 3.3, and the two are equivalent, but only after the Lambert equation is used, and as such the paper has a more developed version of the formula.
  • 3 - The simpler, and easier to understand, method is to use recurance relationship   coupled with   when r=1 shows  . From this equivalence and the generating function definition it logically follows that   (no proof needed here). The real mathematical magic comes from using Lambert's equivalence, this is a result I simply lifted from my corespondance with him (it is also specifically mentioned in "DENSITIES OF THE RANEY DISTRIBUTIONS" arXiv:1211.7259v1 [math.PR] 30 Nov 2012, equation 5).

I am reluctant to change this section back as I do not know who changed it, why they changed it, they may well know far more than me!!! Bbloodaxe (talk) 14:43, 14 January 2014 (UTC) bbloodaxeReply

Modified to keep Gamma ratio comments, and conform with previous logic Bbloodaxe (talk)

Suggestion 3

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I think this article may actually be about Raney numbers... Some of the literature suggests that Fuss-Catalan is given by

 

see Anderson Rational Catalan Combinatorics[1], and articles on Raney numbers too mentioned in article. An alternative is to see N I Fuss work (1791 ish)"Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat", unfortunately I do not speak Latin (but Google does!!). Any ideas on how to resolve??? I have put in reference to this at beginning of article Bbloodaxe (talk) 19:55, 17 January 2014 (UTC) bbloodaxeReply

References