Some Identities of Fully Degenerate Dowling Polynomials and Numbers

. Recently, Kim-Kim introduced the degenerate Whitney numbers of the frst 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.


Introduction
Te defnition and properties of some special polynomials are important, and many academics have studied the defnitions and properties of some polynomials or degenerate polynomials [1][2][3][4].Kim and Kim [5] introduced the degenerate Dowling polynomials and degenerate Whitney numbers of the frst 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 (L, ≤) be a fnite lattice [5,6], which means it is a fnite poset such that every pair x, y of elements in L has a supremum x ∨ y and an infmum x ∧ y.A fnite lattice L is geometric if it is a fnite semimodular lattice which is also atomic.For a fnite geometric lattice L of rank n, Dowling [7] defned the Whitney numbers V L (n, k) of the frst kind and the Whitney numbers W L (n, k) of the second kind.In particular, if L is the Dowling lattices [5][6][7] Q n (G) of rank n over a fnite group G of order m, then the Whitney numbers of the frst kind V Q n (G) (n, k) and the Whitney numbers of the second kind W Q n (G) (n, k) are, respectively, denoted by V m (n, k) and W m (n, k).
From (9), we note that where λ), (n ≥ 1) (see [9]).Te fully degenerate Bell polynomials (see [8,9,13]) are defned by From [13], we have In [9], we note that Te degenerate Stirling numbers of the frst kind and the second kind (see [6,8,9,11,14]) are defned 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.

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.
Te fully degenerate Dowling polynomials are defned by ) are called the fully degenerate Dowling numbers.
Te four identities in the following lemma can be shown just as in Teorem 1, Corollary 1, Teorem 10, and Teorem 17 of [5].

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In particular, when x � 1, we have Theorem 9.For n ≥ 0, Proof.From Teorem 5 and Lemma 3, we note that Discrete Dynamics in Nature and Society From (34), we obtain Teorem 9.

Remarks
Recently, the degenerate Poisson random variable X λ with parameter α > 0, probability mass function is given by (see [15]) Now, we would like to give the relation between the fully degenerate Dowling polynomials and degenerate Poisson random variables.So, we slightly modify our fully degenerate Dowling polynomials which are given by Theorem 13 where X λ is the degenerate Poisson random variable with parameters α/m.
Proof.Let X λ be the degenerate Poisson random variable with parameters α/m, then we have On the other hand, we easily get

Conclusion
In this paper, we introduced the fully degenerate Dowling polynomials D * m,λ (n, x) and the fully degenerate Bell polynomials Bel * n,λ (x), which are degenerate versions of the Dowling polynomials D m (n, x) and the Bell polynomials Bel n (x).
We showed the fully degenerate Dowling polynomials with Whitney numbers in Teorem 5. We used diferent methods for the fully degenerate Dowling polynomials to obtain some identities related to the Stirling numbers of the frst kind and the second kind, the degenerate Stirling numbers of the second kind and the fully degenerate Bell polynomials in Teorems 6-8 and 11.Meanwhile, we investigated the recurrence relations for the fully degenerate Dowling polynomials in Teorem 9.In particular, we let x � 1 and m � 1, and we have also obtained the special relation between the fully degenerate Dowling polynomials and the degenerate Stirling numbers of the second kind in Teorem 10.Furthermore, we used the diferential equation to obtain the identities associated with the fully degenerate Dowling polynomials in Teorem 12. Assume that X λ is the degenerate Poisson random variable with parameter α/m, we showed that the Poisson degenerate central moments E[(mX λ + 1) n,λ ] is equal to D * m,λ (n, α) in Teorem 13.Tis has profound implications for us to continue to study various degenerate versions of many special polynomials and numbers in the future.