Explicit expressions for the related numbers of higher order Appell polynomials

Abstract

In this note, by using the Hasse-Teichmüller 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.

Appell-Carlitz numbers

“…Proof of Theorem 2.10. At this stage, we show that the proof of [12,Theorem 2] which based on the inductive method can also be applied to our situation. Denote by A .…”

“…It may contain constants, variables, certain 'wellknown' operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit." During the recent years, there are many results concerning closed form expressions for special numbers and polynomials in characteristic 0 case, such as Bernoulli, Euler, Cauchy, Apostol-Bernoulli, hypergeometric Bernoulli numbers and polynomials, see [5,6,11,12,14,15,16,19] and the references therein.…”

“…Recently, in order to unify the definitions of several special numbers such as the (higher order) Bernoulli numbers and the (higher order) Cauchy numbers, Hu and Komatsu [12] introduced the concept of the related numbers of higher order Appell polynomials. Their definition is as follows.…”

A new class of higher order hypergeometric Bernoulli polynomials associated with Hermite polynomials

“…N,n (0, 0) is the hypergeometric Bernoulli numbers, (see [8,15]). If we put N = 1, the result reduces to the known result of Pathan and Khan, (see [16]).…”

Generalized hypergeometric Bernoulli numbers

“…In [16], the classical Euler numbers are generalized to define the higher order hypergeometric Euler numbers. Recently, in [12], by using the Hasse-Teichmüller derivatives, two explicit expressions for the related numbers of higher order Appell polynomials are obtained. 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…”