"Eulerian" transform

[Ross La Haye]

FMU, there are 3 Euler transforms, but I'm wondering if there is something
like an "Eulerian" transform known and used, i.e., a transform identical to
the Stirling transform except using the Eulerian numbers (whether of first
or second order) instead of the Stirling numbers of the second kind.  If
not, does anyone think that sequences generated by such a transform would be
of interest...?  For example, let eF(s) denote the "Eulerian" transform of
integer sequence s using the first order Eulerian numbers.  Then for s =
1,1,1,1,1,1,1,1... eF(s) = the factorial numbers, which, of course, equal
the sums of the rows of triangle of first order Eulerian numbers, much like
the Stirling transform of s = the Bell numbers...the sums of the rows of the
triangle of Stirling numbers of the second kind and the binomial transform
of s = the powers of 2...the sums of the rows of Pascal's triangle and so
on...

Ross


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