Numerator polynomials of the Riordan matrices

Abstract: Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. Generalized Euler polynomials ${{g}{n}}\left( x \right)={{\left( 1-x \right)}^{n+1}}\sum\nolimits{m=0}^{\infty }{{{p}{n}}}\left( m \right){{x}^{m}}$, where ${{p}{n}}\left( m \right)$ is the polynomial of degree $\le n$, are the numerator polynomials of the generating functions of diagonals of the ordinary Riordan matrices. Generalized Narayana polynomials ${{h}{n}}\left( x \right)={{\left( 1-x \right)}^{2n+1}}\sum\nolimits{m=0}^{\infty }{\left( m+1 \right)...\left( m+n \right){{p}{n}}}\left( m \right){{x}^{m}}$ are the numerator polynomials of the generating functions of diagonals of the exponential Riordan matrices. In paper, the properties of these two types of numerator polynomials and the constructive relationships between them are considered. Separate attention is paid to the numerator polynomials of Riordan matrices associated with the family of series ${\left( \beta \right)}a\left( x \right)=a\left( x{}_{\left( \beta \right)}{{a}^{\beta }}\left( x \right) \right)$.