Riordan arrays and generalized Euler polynomials

TL;DR

This paper explores generalized Euler polynomials through the lens of Riordan arrays, revealing new transformations and connections with combinatorial objects like Stirling numbers and binomial sequences.

Contribution

It introduces a novel perspective by linking generalized Euler polynomials to Riordan arrays, highlighting transformations involving key combinatorial objects.

Findings

Establishes a connection between generalized Euler polynomials and Riordan arrays.

Demonstrates transformations involving binomial sequences, Stirling numbers, and multinomial coefficients.

Shows the constructiveness of the Riordan array approach in studying these polynomials.

Abstract

Generalization of the Euler polynomials ${{A}_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{m}^{n}}{{x}^{m}}}$ are the polynomials ${{\alpha }_{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits_{m=0}^{\infty }{{{u}_{n}}}\left( m \right){{x}^{m}}$, where ${{u}_{n}}\left( x \right)$ is the polynomial of degree $n$. These polynomials appear in various fields of mathematics, which causes a variety of methods for their study. In present paper we will consider generalized Euler polynomials as an attribute of the theory of Riordan arrays. From this point of view, we will consider the transformations associated with them, with a participation of such objects as binomial sequences, Stirling numbers, multinomial coefficients, shift operator, and demonstrate a constructiveness of the chosen point of view.