Abstract

Recently, Kim-Kim introduced the degenerate Whitney numbers of the first and second kind involving the degenerate Dowling polynomials and numbers. In this paper, we introduce the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and derive some identities involving these polynomials. We also obtain the generating functions, expressions, and recurrence relations for the fully degenerate Dowling polynomials and the fully degenerate Bell polynomials and other special polynomials and numbers.

1. Introduction

The definition and properties of some special polynomials are important, and many academics have studied the definitions and properties of some polynomials or degenerate polynomials [1–4]. Kim and Kim [5] introduced the degenerate Dowling polynomials and degenerate Whitney numbers of the first and second kind and obtain some explicit identities. Kim and Kim [6] introduced degenerate r-Dowling polynomials related to the degenerate r-Whitney numbers of the second kind.

Let be a finite lattice [5, 6], which means it is a finite poset such that every pair of elements in has a supremum and an infimum . A finite lattice is geometric if it is a finite semimodular lattice which is also atomic. For a finite geometric lattice of rank , Dowling [7] defined the Whitney numbers of the first kind and the Whitney numbers of the second kind. In particular, if is the Dowling lattices [5–7] of rank over a finite group of order , then the Whitney numbers of the first kind and the Whitney numbers of the second kind are, respectively, denoted by and .

At the first, we give some definitions and identities needed in this paper. For any nonzero , the degenerate exponential functions are defined bywhere (see [5, 6, 8–11]).

The degenerate Whitney numbers and are defined by the following generating functions with parallel structures,where (see [5, 6]).

Equivalently, relations (2) and (3) are given by

For , the degenerate Dowling polynomials (see [5, 11]) are given bywhen , are the degenerate Dowling numbers. Note thatwhere are the Dowling polynomials.

From (6), we have(see [6, 11]).

The degenerate Bell polynomials (see [5, 8–12]) are defined bywhen , are the degenerate Bell numbers.

From (9), we note thatwhere , (see [9]).

The fully degenerate Bell polynomials (see [8, 9, 13]) are defined by

From [13], we have

In [9], we note that

The degenerate Stirling numbers of the first kind and the second kind (see [6, 8, 9, 11, 14]) are defined by

By the inversion of (14) and (15), we have

Recently, we are interested in the degenerate Dowling polynomials and numbers and the degenerate Bell polynomials and numbers. In this paper, we study the fully degenerate Dowling polynomials and numbers and the fully degenerate Bell polynomials and numbers. Meanwhile, we derive some identities and expressions of them.

2. Fully Degenerate Dowling Polynomials

In this section, we introduce the fully degenerate Dowling polynomials. We also show several identities and properties related to the fully degenerate Dowling polynomials and numbers.

The fully degenerate Dowling polynomials are defined bywhen , are called the fully degenerate Dowling numbers.

The four identities in the following lemma can be shown just as in Theorem 1, Corollary 1, Theorem 10, and Theorem 17 of [5].

Lemma 1. For ,

Lemma 2. For with ,

Lemma 3. For ,

Lemma 4. For with ,

Theorem 5. For , we have

Proof. By Lemma 1 and (18), we haveSo, from (24), we obtain Theorem 5.

Theorem 6. For , we have

Proof. From (18), we haveFrom (26), we obtain Theorem 6.

In particular, when , we note that

Theorem 7. For , we have

Proof. From (18), we getSo, by (29), we obtain Theorem 7.

Theorem 8. For , we have

Proof. By (18), we getFrom (31), we get Theorem 8.

In particular, when , we have

Theorem 9. For ,

Proof. From Theorem 5 and Lemma 3, we note thatFrom (34), we obtain Theorem 9.

Theorem 10. For ,

Proof. By Theorem 5 and Lemma 2, let , we haveBy (36), we obtain Theorem 10.