The purpose of this article is to construct generating functions for new families of special polynomials including two parametric kinds of Eulerian-type polynomials. Some fundamental properties of these functions are given. By using these generating functions and the Euler’s formula, some identities and relations among trigonometric functions, two parametric kinds of Eulerian-type polynomials, Apostol-type polynomials, the Stirling numbers and Fubini-type polynomials are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are given.

The aim of this paper is to construct generating functions for a new family of polynomials, which are called parametric Hermite-based Milne-Thomson type polynomials. Many properties of these polynomials with their generating functions are investigated. These generating functions give us generalization of some well-known generating functions for special polynomials such as Hermite type polynomials, Milne-Thomson type polynomials, and Apostol type polynomials. Using the Euler formula, functional equation method for generating function, and differential operator technique, we give relations among parametric Hermite-based Milne-Thomson type polynomials, the Bernoulli numbers, the Euler numbers, the Chebyshev polynomials, the Bernstein basis functions, homogeneous harmonic polynomials, and parametric kinds of Apostol type polynomials. Moreover, some computational formulas for these polynomials are derived. Finally, using Wolfram Mathematica version 12.0, some of these polynomials and their generating functions are illustrated by their plots under the special conditions. Potential relationships and connections of this paper's results with the results of previous and future research are pointed out.

The Fubini type polynomials have many application not only especially in combinatorial analysis, but also other branches of mathematics, in engineering and related areas. Therefore, by using the p-adic integrals method and functional equation of the generating functions for Fubini type polynomials and numbers, we derive various different new identities, relations and formulas including well-known numbers and polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers of the second kind, the λ-array polynomials and the Lah numbers.

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