Explicit expressions for the related numbers of higher order Appell polynomials

TL;DR

This paper derives explicit formulas, including a determinant expression, for the related numbers of higher order Appell polynomials using Hasse-Teichmüller derivatives, connecting them to classical special numbers.

Contribution

It introduces new explicit and determinant formulas for higher order Appell polynomial related numbers, linking them to classical special numbers.

Findings

Derived explicit formulas for related numbers of higher order Appell polynomials.

Presented a determinant expression involving hypergeometric Bernoulli and Cauchy numbers.

Connected new formulas to classical determinant expressions from Glaisher's 1875 work.

Abstract

In this note, by using the Hasse-Teichm\"uller derivatives, we obtain two explicit expressions for the related numbers of higher order Appell polynomials. One of them presents a determinant expression for the related numbers of higher order Appell polynomials, which involves several determinant expressions of special numbers, such as the higher order generalized hypergeometric Bernoulli and Cauchy numbers, thus recovers the classical determinant expressions of Bernoulli and Cauchy numbers stated in an article by Glaisher in 1875.