Mixed-Type Hypergeometric Bernoulli-Gegenbauer Polynomials

In this paper, we consider a 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 focus our attention on some algebraic and differential properties of this class of polynomials, including its explicit expressions, derivative formulas, matrix representations, matrix-inversion formulas, and other relations connecting it with the hypergeometric Bernoulli polynomials. Furthermore, we show that unlike the hypergeometric Bernoulli polynomials and Gegenbauer polynomials, the mixed-type hypergeometric Bernoulli-Gegenbauer polynomials do not fulfill either Hanh or Appell conditions.

Automated summary

This paper investigates a novel family of mixed-type hypergeometric Bernoulli-Gegenbauer polynomials, exploring their algebraic and differential properties, and their relationships with hypergeometric Bernoulli polynomials. It is found that these polynomials do not fulfill Hanh or Appell conditions.