Eulerian Polynomials: From Euler’s Time to the Present

Abstract

Abstract. 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.

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