Mixed-type hypergeometric Bernoulli-Gegenbauer polynomials: some properties (Q6651486)

The purpose of this article is to study mixed-type hypergeometric Bernoulli-Gegenbauer polynomials by using the generating function. Some well-known notations for number systems are given. It is not necessary to state that \(0^0=1\), since this is a standard convention. The corresponding differential equation for hypergeometric Bernoulli-Gegenbauer polynomials is referred to. The standard generating function for Gegenbauer polynomials which occurs after formula (10) can be found in \textit{E. D. Rainville}'s book [Special functions. Bronx, N. Y.: Chelsea Publishing Company (1971; Zbl 0231.33001), p. 277], and not in the cited [\textit{G. Szegö}, Orthogonal polynomials. 4th ed. Providence, RI: American Mathematical Society (AMS) (1975; Zbl 0305.42011)]. On the other hand, the generating functions for the monic Gegenbauer polynomials (9) and (10) are not proved. The orthogonality relation for Gegenbauer polynomials with a new weight function on page 5 cannot be found in [{G. Szegö}, loc. cit.; Chapter IV]. Then diagrams for the zeros of the new polynomials are given. I have tried to correct some of the statements in the paper. To sum up, it would have been better to follow {E. D. Rainville}'s book [loc. cit.] and his notation for Gegenbauer polynomials.