Several formulae involving the Cauchy polynomials, the Bernoulli polynomials, and the generalized Stirling numbers (2401.13696v3)

Abstract: In this paper, we obtain new relationships between the Cauchy polynomials, the Bernoulli polynomials, and the generalized (or weighted) Stirling numbers of both kinds. Furthermore, we derive some identities involving the Cauchy polynomials, the Daehee polynomials, and the hyperharmonic polynomials. In addition, we consider the poly-Cauchy polynomials and the weighted type poly-Bernoulli polynomials and retrieve some of their relationships in the light of our previous findings. Finally, we give various relationships involving the rational values of the Cauchy polynomials and those of the Bernoulli polynomials in terms of $r$-Whitney numbers.