In this paper, we investigate the recently introduced degenerate Bernstein polynomials and operators and derive some of their properties. Also, we give some properties of the degenerate Euler numbers and polynomials and their connection with the degenerate Euler polynomials.

In this paper, we define new kind of Daehee numbers, the degenerate Daehee numbers of the third kind, using degenerate log function as generating function. We obtain some identities for the degenerate Daehee numbers of the third kind associated with the Daehee, degenerate Daehee and degenerate Daehee numbers of the second kind. In addition, we derive a differential equation associated with the degenerate log function. We deduce some identities from the differential equation.

In this paper, we study sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials and represent each of them in terms of Chebyshev polynomials of all kinds. Here the coefficients involve terminating hypergeometric functions ₂F₁ and these representations are obtained by explicit computations.

This paper treats the connection problem of expressing sums of finite products of Chebyshev polynomials of the third and fourth kinds in terms of five classical orthogonal polynomials. In fact, by carrying out explicit computations each of them are expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer, and Jacobi polynomials which involve some terminating hypergeometric functions ${}_2 F_0, {}_2 F_1$, and ${}_3 F_2$.

This paper is concerned with representing sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials in terms of several classical orthogonal polynomials. Indeed, by explicit computations each of them is expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials which involve the hypergeometric functions ${}_1 F_1$ and ${}_2 F_1$.

In this paper, we represent the generating function of w-torsion Fubini polynomials by fermionic p-adic integral on Z_p. Then we consider a quotient of such fermionic p-adic integrals on Z_p, representing generating functions of three w-torsion Fubini polynomials and derive some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials.

In this paper, we introduce two discrete degenerate random variables, namely the degenerate binomial and degenerate Poisson random variables. We deduce the expectations of the degenerate binomial random variables. We compute the generating function of the moments of the degenerate Poisson random variables, which leads us to define the new type degenerate Bell polynomials, and hence obtain explicit expressions for the moments of those random variables in terms of such polynomials. We also get the variances of the degenerate Poisson random variables. Finally, we illustrate two examples of the degenerate Poisson random variables.