Eulerian summation operators and a remarkable family of polynomials

TL;DR

This paper explores polynomial families connected through Eulerian summation operators, revealing their combinatorial properties, symmetries, and identities related to super Catalan numbers, with implications for integer-valuedness and recursive structures.

Contribution

It introduces new polynomial families linked by Eulerian operators and uncovers their combinatorial and symmetry properties, including identities for super Catalan numbers.

Findings

Polynomials are integer-valued at integral points.

Identities involving super Catalan numbers are derived.

Recursion formulas for values at half-integral points are established.

Abstract

We give several families of polynomials which are related by Eulerian summation operators. They satisfy interesting combinatorial properties like being integer-valued at integral points. This involves nearby-symmetries and a recursion for the values at half-integral points. We also obtain identities for super Catalan numbers.