Let Y be a random variable such that the moment generating function of Y exists in a neighborhood of the origin. The aim of this paper is to study probabilistic versions of the degenerate Fubini polynomials and the degenerate Fubini polynomials of order r, namely the probabilisitc degenerate Fubini polynomials associated with Y and the probabilistic degenerate Fubini polynomials of order r associated with Y. We derive some properties, explicit expressions, certain identities and recurrence relations for those polynomials. As special cases of Y, we treat the gamma random variable with parameters $$\alpha ,\beta > 0$$ α , β > 0 , the Poisson random variable with parameter $$\alpha > 0$$ α > 0 , and the Bernoulli random variable with probability of success p.

The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using this new sequence, the properties of the power series, and the combinatorial methods.

We use the elementary and analytic methods and the properties of Chebyshev polynomials to study the computational problem of the reciprocal sums of one‐kind Chebyshev polynomials and give several interesting identities for them. At the same time, we also give a general computational method for this kind of reciprocal sums.

Spivey found a remarkable recurrence relation for Bell numbers, which was generalized to that for Bell polynomials by Gould-Quaintance. The aim of this article is to generalize their recurrence relation for Bell polynomials to that for the probabilistic Dowling polynomials associated with Y and also that for the probabilistic r -Dowling polynomials associated with Y . Here Y is a random variable whose moment generating function exists in a neighborhood of the origin.

The main aim of this paper is to use the analytic methods and the properties of the classical Gauss sums to research the computational problem of one kind hybrid power mean containing the character sums of polynomials and two-term exponential sums modulo p, an odd prime, and acquire several accurate asymptotic formulas for them.