By Doron Zeilberger
I thank Christophe for his elegant remark, and remark in my turn that a trivial extension of
my first proof (replace -1 by t) (x is already taken in the paper) yields the famous connection
formula that he used.
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Written: June 26, 2011.
This quickie was written after
Dan Haran
sent me the following
question .
Added Jan. 18, 2012:
Christophe Vignat commented that a yet simpler version of the first simple proof of the fact that
(-1)n = Σ (-1)k k! S(n,k)
is a direct consequence of the fact that the Stirling numbers of the second kind are the connection coefficients between the
monomials xn and the descending factorials (x)j = x(x-1)...(x-j+1). In other words
xn = Σ S(n,k) (x)k .
Taking x=-1 yields the identity.
Personal Journal of Shalosh B. Ekhad and Doron Zeilberger