Riordan arrays and generalized Euler 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.