Exponential Riordan arrays and generalized Narayana polynomials

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.