[seqfan] Re: LambertW Identity - Conjecture

Paul D Hanna pauldhanna at juno.com
Mon Mar 16 14:59:25 CET 2009

Seqfans,
Below I give a complementary identity.

>> Note that if V=1, then the above claim reduces to:
>> x = W(x*exp(x)).
>
> You agree that this is the definition of the W function ?
> (Just to make sure what we consider as definition and what as result...)
>
> Maximilian

Yes, of course (I should have been more precise).
The point is that the identity reduces to the expected result at the trivial case V=1.

Starting from the alternate expression of the definition:
x = W(x)*exp(W(x))
we get the following complementary identity.

-------------------------------------------------------
Conjecture II.
Given V is an arbitrary real and J is an integer, then

V^(J^2)*x = Sum_{n>=0} x^n/n! * Sum_{k>=1} V^((n+k+J-1)^2)*(-1)^(k-1)*(n+1)*(n+k)^(k-2)/(k-1)!*x^k
-------------------------------------------------------

Note that at V=1 this statement reduces to:
x = W(x)*exp(W(x))
as expected by definition of Lambert W(x).

EXAMPLE using PARI code:
V=2;J=3;
sum(n=0,100,x^n/n!*sum(k=1,100,V^((n+k+J-1)^2)*(-1)^(k-1)*(n+1)*(n+k)^(k-2)/(k-1)!*x^k) )+O(x^100)

RETURNS: 512*x + O(x^100)

Regards,
Paul

---------- Original Message ----------
From: "Paul D Hanna" <pauldhanna at juno.com>
To: seqfan at list.seqfan.eu
Subject: [seqfan]  LambertW Identity - Conjecture
Date: Mon, 16 Mar 2009 06:59:37 GMT

Seqfans,
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.
----------------------------------------------------------------
Conjecture:
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:
V=2;J=3;
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.