[seqfan] LambertW Identity - Conjecture

Paul D Hanna pauldhanna at juno.com
Mon Mar 16 07:59:37 CET 2009

       Here is an identity that I just found related to the LambertW function. 
It is a conjecture and I need a proof. 
The statement is a variant of the sum in A155200:  
  exp( Sum_{n>=1} 2^(n^2) * x^n/n )  = 1+2x+10x^2+188x^3+... 
in which I attempted to replace exp(x) with  
  W(x) = Sum_{n>=1} (-n)^(n-1)*x^n/n!
Here is the surprising result of this line of inquiry.  
Given V is any real variable and J is an integer, then 
V^(J^2)*x = Sum_{n>=1} (-n)^(n-1)/n!*x^n*[Sum_{k>=0} V^((n+k+J-1)^2)*n^k*x^k/k!]. 
Note that if V=1, then the above claim reduces to:
x = W(x*exp(x)).
EXAMPLE using PARI code:
sum(n=1,100,(-n)^(n-1)/n!*x^n*sum(k=0,100,V^((k+n+J-1)^2)*n^k*x^k/k! +O(x^100)))
RETURNS:  512*x + O(x^101)
Besides a proof, I am mostly interested in what this is telling us about what is really going on in sums like that given in A155200 and above. 
I welcome any comments. 
(dedicated to Vladeta Jovovic).

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