Talk:Fuss–Catalan number

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 (talk • contribs)

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

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

:$B\_\{p,r\}(z):=\backslash sum\_\{m=0\}^\backslash infty\; A\_m(p,r)z^m$, then the Wojciech Mlotkowski paper (see references), shows that as

:$A\_\{m+1\}(p,1)=\; A\_\{m\}(p,p)$ then it directly follows that $B\_\{p,1\}(z)\; =\; 1+zB\_\{p,p\}(z)$.

This can extended by using Lambert's equivalence $B\_\{p,1\}(z)^r=B\_\{p,r\}(z)$ to the general generating function, for all the Fuss-Catalan numbers:

:$B\_\{p,r\}(z)\; =\; [1+zB\_\{p,r\}(z)^\{p/r\}]^r$.

to

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

:$B\_\{p,r\}(z):=\backslash sum\_\{m=0\}^\backslash infty\; A\_m(p,r)z^m$.

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

:$A\_m(p,p)=A\_\{m+1\}(p,1)$.

Then the Wojciech Mlotkowski paper (see references), shows that $B\_\{p,1\}(z)\; =\; 1+zB\_\{p,p\}(z)$.

This can extended by using Lambert's equivalence $B\_\{p,1\}(z)^r=B\_\{p,r\}(z)$ to:

:$B\_\{p,r\}(z)\; =\; [1+zB\_\{p,r\}(z)^\{p/r\}]^r$.

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 $A\_m(p,p)=A\_\{m+1\}(p,1)$, 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 $B\_\{p,1\}(z)\; =\; 1+zB\_\{p,p\}(z)$. This is a a logical consequence that $A\_m(p,p)=A\_\{m+1\}(p,1)$, and merely arrived at by substitution into the generating formula $B\_\{p,r\}(z):=\backslash sum\_\{m=0\}^\backslash infty\; A\_m(p,r)z^m$. I will agree that the paper shows a similar formula, namely $B\_\{p,1\}(z)\; =\; 1+zB\_\{p,1\}(z)^p$ 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 $A\_m(p,r)=A\_m(p,r-1)+A\_\{m-1\}(p,p+r-1)$ coupled with $A\_m(p,0)=0$ when r=1 shows $A\_m(p,p)=A\_\{m+1\}(p,1)$. From this equivalence and the generating function definition it logically follows that $B\_\{p,1\}(z)\; =\; 1+zB\_\{p,p\}(z)$ (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) bbloodaxe

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

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

:$A\_m(p)\; =\; \backslash frac\{1\}\{mp+1\}\backslash binom\{mp+1\}\{m\}$

see Anderson Rational Catalan Combinatoricshttp://www.math.miami.edu/~armstrong/RCC_AIM.pdf, 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) bbloodaxe

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