  The aim of this paper is to construct generating functions for new families
of special polynomials including the Appel polynomials, the Hermite-Kamp\`e de
F\`eriet polynomials, the Milne-Thomson type polynomials, parametric kinds of
Apostol type numbers and polynomials. Using Euler's formula, relations among
special functions, Hermite-type polynomials, the Chebyshev polynomials and the
Dickson polynomials are given. Using generating functions and their functional
equations, various formulas and identities are given. With help of
computational formula for new families of special polynomials, some of their
numerical values are given. Using hypegeometric series, trigonometric functions
and the Euler's formula, some applications related to Hermite-type polynomials
are presented. Finally, further remarks, observations and comments about
generating functions for new families of special polynomials are given.
