© 2022, The Author(s) under exclusive licence to The Royal Academy of Sciences, Madrid.With aid of generating functions and their functional equation methods and special functions involving trigonometric functions, the motivation of this paper is to study by blending certain families polynomials associated with the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Hermite type polynomials, the Stirling numbers, the telephone numbers, the Chebyshev polynomials. Therefore, the purpose of this paper is to examine certain families of numbers and functions related to generalized Hermite–Kampè de Fèriet polynomials and trigonometric functions. By using functional equations of generating functions, we derive numerous new formulae and relations involving parametric Hermite type polynomials, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the generalized Hermite–Kampè de Fèriet polynomials and the telephone numbers. Moreover, applying derivative operator to these generating functions, we give many recurrence relations and computational formulae, and certain finite sums. Finally, some special cases of these results are reduced to not only the well-known Chebyshev polynomials, which have applications in a wide variety of different areas, but also special trigonometric functions and finite sums.