New Eulerian numbers of type D

TL;DR

This paper introduces a new array of type D Eulerian numbers, providing recurrence relations, formulas, and generating functions, and explores related probability distributions with moments linked to Eulerian polynomials of types A, B, and D.

Contribution

It presents a novel array of type D Eulerian numbers with new recurrence relations, formulas, and connections to probability distributions, expanding the understanding of Eulerian number types.

Findings

Derived a new recurrence relation for type D Eulerian numbers.

Established the Worpitzky formula and generating function for the new array.

Connected probability distributions to Eulerian polynomials of types A, B, and D.

Abstract

We introduce a new array of type $D$ Eulerian numbers, different from that studied by Brenti, Chow and Hyatt. We find in particular the recurrence relation, Worpitzky formula and the generating function. We also find the probability distributions whose moments are Eulerian polynomials of type $A$, $B$ and $D$.