[seqfan] Re: LambertW Identity - Conjecture
Paul D Hanna
pauldhanna at juno.com
Mon Mar 16 23:15:49 CET 2009
Seqfans,
This is one of those identities that on the surface look deeper than they really are:
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!].
If we form an array of coefficients
T(n,k) = V^((n+k+J-1)^2)*(-n)^(n-1)/n! * n^k/k!
and sum on antidiagonals,
the terms cancel everywhere except the upper left corner,
leaving the value V^(J^2).
So, sorry for the false call!
Paul
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