Mixed-type hypergeometric Bernoulli-Gegenbauer polynomials: some properties

We consider the novel family of the mixed-type hypergeometric Bernoulli-Gegenbauer polynomials. This family represents a fascinating fusion between two distinct categories of special functions: hypergeometric Bernoulli polynomials and Gegenbauer polynomials. We collect some recent results concerning algebraic and differential properties of this class of polynomials and use some them to deduce an ordinary differential equation satisfied by these polynomials. Some numerical illustrative examples about the behavior of the zeros of these polynomials are given.