Explicit expressions for the related numbers of higher order appell polynomials

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.

Mathematics Subject Classification (2010): 11B68, 11B37, 11C20, 15A15, 33C15.

Keywords: Appell polynomials, Bernoulli numbers, hypergeometric Bernoulli numbers, hypergeometric functions, explicit expressions, determinants, recurrence relations