The main object of this paper is to construct a new generating function of the (q‐) Bernstein‐type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and derivative of the (q‐) Bernstein‐type polynomials. We also give relations between the (q‐) Bernstein‐type polynomials, Hermite polynomials, Bernoulli polynomials of higher order, and the second‐kind Stirling numbers. By applying Mellin transformation to this generating function, we define interpolation of the (q‐) Bernstein‐type polynomials. Moreover, we give some applications and questions on approximations of (q‐) Bernstein‐type polynomials, moments of some distributions in Statistics.

Recently, Sofonea (Gen. Math. 16:47-54, 2008) considered some relations in the context of quantum calculus associated with the q-derivative operator D q $D_{q}$ and divided difference. As applications of the post-quantum calculus known as the ( p , q ) $(p,q)$ -calculus, we derive several relations involving the ( p , q ) $(p,q)$ -derivative operator and divided differences.

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials.

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order $\alpha$ and investigate multifarious correlations and formulas including some summation formulas and derivative properties. Also, we acquire diverse implicit summation formulas and symmetric identities for Bell-based Bernoulli polynomials of order $\alpha$. Moreover, we attain several interesting formulas of Bell-based Bernoulli polynomials of order $\alpha$ arising from umbral calculus.