Exponential Riordan arrays and generalized Narayana polynomials

TL;DR

This paper explores the relationship between generalized Euler and Narayana polynomials, which are connected to the diagonals of Riordan arrays, revealing new insights into their structural connections.

Contribution

It establishes a constructive relationship between numerator polynomials of diagonals of ordinary and exponential Riordan arrays, linking generalized Euler and Narayana polynomials.

Findings

Identifies the connection between generalized Euler and Narayana polynomials.

Provides a constructive method relating these polynomials.

Enhances understanding of Riordan array diagonal generating functions.

Abstract

Generalized Euler polynomials ${{\alpha }_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{p}_{n}}}\left( m \right){{x}^{m}}$, where ${{p}_{n}}\left( x \right)$ is the polynomial of degree $n$, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan arrays. Generalized Narayana polynomials ${{\varphi }_{n}}\left( x \right)={{\left( 1-x \right)}^{2n+1}}\sum\nolimits_{m=0}^{\infty }{\left( m+1 \right)...\left( m+n \right){{p}_{n}}}\left( m \right){{x}^{m}}$ are the numerator polynomials of the generating functions of diagonals of the exponential Riordan arrays. In present paper we consider the constructive relationship between these two types of numerator polynomials.