[seqfan] Re: A Remarkable Eigenfunction - A179497

[Paul D Hanna]

SeqFans, 
      Sorry, typo's need to be corrected! 
The specific case of t=3 should be stated as: 
(*) [G_{n+1}(x)]^3 = G(x)^3 * G_n'(x) for all n 
 
The general case should have been written: 
(**) [G_{n+1}(x)]^t = G(x)^t * G_n'(x) for all n 
where t is an implicit parameter in G(x). 
 
---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: A Remarkable Eigenfunction - A179497
Date: Mon, 2 Aug 2010 15:18:40 GMT

SeqFans,
     I just realized that these results can be generalized.
Suppose G(x) satisfies:
(*)  G_n(x)^3 = G(x)^3 * G_n'(x) for all n 
where G_n(x) = n-th iteration of G(x) and G(0)=0, G'(0)=1, 
then G = G(x) satisfies: 
  G = x + G^3 + G^3*D(G^3)/2! + G^3*D(G^3*D(G^3))/3! + G^3*D(G^3*D(G^3*D(G^3)))/4! + ...
and   
  G_n(x) = x + n*G^3 + n^2*G^3*D(G^3)/2! + n^3*G^3*D(G^3*D(G^3))/3! + n^4*G^3*D(G^3*D(G^3*D(G^3)))/4! + ...
 
Explicitly,
  G(x) = x + 6*x^3/3! + 540*x^5/5! + 156240*x^7/7! + 96480720*x^9/9! + 104661849600*x^11/11! + 177947471782080*x^13/13! + 439942718370355200*x^15/15! +...

(I will submit this sequence of coefficeints to OEIS tonight sometime.) 

The obvious generalization is as follows.
if 
(**) G_n(x)^t = G(x)^t * G_n'(x) 
then G=G(x) satisfies:
   G_n(x) = x + n*G^t + n^2*G^t*D(G^t)/2! + n^3*G^t*D(G^t*D(G^t))/3! + n^4*G^t*D(G^t*D(G^t*D(G^t)))/4! + ...

I find this to be an interesting aside. 

I would like to see some nice formula for the coefficients in these series. 
 
Thanks, 
   Paul 

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