Invited address at the 10-th Annual Ulam Colloquium,
University of Florida, Gainesville, February 18, 2008. Published in
"The Legacy of Alladi Ramakrishnan in the Mathematical Sciences,"
K. Alladi, J.R. Klauder, C.R. Rao, eds., Springer, New York Dordrecht
Heidelberg London, 2010, p. 253-273.
EULERIAN POLYNOMIALS: FROM EULER'S TIME TO THE PRESENT
The polynomials commonly called "Eulerian" today have
been introduced by Euler himself in his famous book
"Institutiones calculi differentialis cum eius usu in
analysi finitorum ac Doctrina serierum" (chap. VII), back
in 1755. They have been since thoroughly studied,
extended, applied. The purpose of the present paper is to
go back to Euler's memoir, find out his motivation and
reproduce his derivation, surprisingly partially forgotten.
The rebirth of those polynomials in a q-environment is
due to Carlitz two centuries after Euler. A brief overview
of Carlitz's method is given, as well as a short
presentation of combinatorial works dealing with
natural extensions of the classical Eulerian polynomials.
foata at math dot u-strasbg dot fr
The following version is available: