  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 $_{\left( \beta \right)}a\left( x \right)=a\left( x{}_{\left(
\beta \right)}{{a}^{\beta }}\left( x \right) \right)$.
