Generalized Eulerian Triangles and Some Special Production Matrices

TL;DR

This paper introduces generalized Eulerian triangles derived from special production matrices, linking them to orthogonal polynomial moments and Catalan-based triangles using advanced transforms and Riordan arrays.

Contribution

It presents a novel framework connecting generalized Eulerian triangles with orthogonal polynomial moments and Catalan structures via the Sumudu transform and Riordan arrays.

Findings

Generalized Eulerian triangles are coefficient arrays of polynomial moments.

These triangles relate to Catalan generating functions through the  transform.

The approach employs Sumudu transforms, Jacobi continued fractions, and Riordan arrays.

Abstract

We show how some special production matrices may be used to define families of generalized Eulerian triangles. We furthermore show that these generalized Eulerian triangles are the coefficient arrays of polynomials which are the moments of families of orthogonal polynomials. Using the previously defined $\mathcal{T}$ transform, we associate these generalized Eulerian triangles to triangles defined by Catalan generating functions. Again, these new triangles are the coefficient arrays of polynomial moment sequences. The main tools used are the Sumudu transform, Jacobi continued fractions and Riordan arrays.