Computer Modeling in Engineering & Sciences |

DOI: 10.32604/cmes.2022.022103

ARTICLE

Some Properties of Degenerate r-Dowling Polynomials and Numbers of the Second Kind

1Department of Mathematics Education, Daegu Catholic University, Gyeongsan, 38430, Korea

2School of Electronic and Electric Engineering, Daegu University, Gyeongsan, 38453, Korea

*Corresponding Author: Hye Kyung Kim. Email: hkkim@cu.ac.kr

Received: 21 February 2022; Accepted: 11 May 2022

Abstract: The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics. In recent years, some mathematicians have studied degenerate version of them and obtained many interesting results. With this in mind, in this paper, we introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind. We derive many interesting properties and identities for them including generating functions, Dobinski-like formula, integral representations, recurrence relations, differential equation and various explicit expressions. In addition, we explore some expressions for them that can be derived from repeated applications of certain operators to the exponential functions, the derivatives of them and some identities involving them.

Keywords: Dowling lattice; Whitney numbers and polynomials; r-Whitney numbers and polynomials of the second kind; r-Bell polynomials; r-Stirling numbers; dowling numbers and polynomials

Mathematics Subject Classification: 11F20; 11B68; 11B83

The Stirling number

When

Dowling [2] constructed a certain lattice for a finite group of order m, called Dowling lattice, and using the Möbius function, he introduced the corresponding Whitney numbers of the first kind

For

As a generalization of the the Whitney numbers

and

respectively, for

When

We note that

Note that the

The r-Whitey numbers of both kinds and r-Dowling polynomials were studied by several authors. The references [2–5,7–15] provided readers more information. In particular, Cheon et al. [8] and Corcino et al. [11] gave combinatorial interpretations of the r-Whitney numbers of the first and second kind, respectively. In recently years, many mathematicians have been studied the degenerate special polynomials and numbers, and have obtained many interesting results [14,16–24]. In particular, the generating functions of (degenerate) special numbers and polynomials have various applications in many fields as well as mathematics and physics [1–32]. Kim et al. [14] introduced the degenerate Whitney numbers of the first kind and the second kind of Dowling lattice

and

With these in mind, we naturally introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers

In this section, we introduce the basic definitions and properties of the degenerate r-Dowling polynomials and numbers needed in this paper.

For

Cheon et al. [8] introduced the r-Dowling polynomials associated with the r-Whitney numbers

By (1) and (4), the generating function of r-Dowling polynomials is given by

where

Corcino et al. [11] studied asymptotic formulas for r-Whitney numbers of the second kind with integer and real parameters. They also obtained the range of validity of each formula.

As is well known, for any

where

The degenerate Stirling numbers of the second kind are given by

Kim et al. studied the unsigned degenerate r-Stirling numbers of the second kind defined by

From (7), the generating function of the degenerate r-Stirling numbers of the second kind is given by

where j is a non-negative integer.

In view of (8), the degenerate r-Bell polynomials are given by

From (9), it is easy to show that the generating function of degenerate r-Bell polynomials is given by

when

Kim et al. introduced the

From (11), we easily get

From (12), we note that

3 Degenerate r-Dowling Polynomials and Numbers

In this section, we explore various properties for the degenerate r-Dowling polynomials and numbers.

From (1), the degenerate r-Whitney numbers

Lemma 3.1. [14] For

In Lemma 3.1, when

From Lemma 3.1, (6) and (8), we get

The next theorem is a recurrence relation of the degenerate Whitney numbers of the second kind.

Theorem 3.1. For

Proof. From (14), we observe that

By comparing the coefficients of both sides of (16), we get the desired recurrence relation.

The following theorem shows that the degenerate r-Whitney numbers of second kind expresses the finite sum of degenerate falling factorials.

Theorem 3.2. For

Proof. By (5) and Lemma 3.1, we observe that

By comparing the coefficients of both sides of (17), we get the desired result.

In Theorem 3.2, when

In this paper, we naturally consider the degenerate

When

When

When

Theorem 3.3. For

Proof. From Lemma 3.1 and (18), we observe that

By (19), we have the generating function of degenerate r-Dowling polynomials of the second kind.

When

When

Theorem 3.4. (Dobinski-like formula)

For

When

Proof. From (5) and Theorem 3.3, we note that

By comparing the coefficients of both sides of (20), we have Dobinski-like formula for the degenerate r-Dowling polynomials.

In the following theorem and corollary, we have integral representations of the degenerate r-Whitney numbers and the degenerate r-Dowling polynomials, respectively.

Theorem 3.5. For

where

Proof. From Lemma 3.1, we get

Therefore, by (21) we have the desired result.

Corollary 3.1. For

Proof. By Lemma 3.1 and Theorem 3.5, we have

From (22), we get the desired identity.

Lemma 3.2. For

Proof. From Theorem 3.2 and (13), we get

By (23), we obtain the desired result.

The next theorem is a recurrence relation of degenerate r-Dowling polynomials.

Theorem 3.6. For

Proof. From (18) and Lemma 3.2, we have

Here

Theorem 3.7. For

Proof. For

By comparing the coefficients of both sides of (25), we get what we want.

The following theorem is another recurrence relation of degenerate r-Dowling polynomials.

Theorem 3.8. For

Proof. From Theorem 3.3, we note that

On the other hand, by (26), we get

By comparing the coefficients of (26) with (27), we get the desired identity.

Remark. When

Next, we explore two identities including degenerate r-Dowling polynomials that can be derived from repeated applications of certain operators to the degenerate exponential functions.

Theorem 3.9. For

Proof. First, we observe that

By (29) and Theorem 3.4,

From (30), we have what we want.

Let

By Theorem 3.3, we have

From (31), the generating function of

Theorem 3.10. For

Proof. Let

By (33), we get

By (34), we attain the desired result.

Remark. When

In Theorem 3.1, when

From (35), we obtain

In (35), when

From (36),

In (35), when

From (37), we get

Thus, by (38), we have

In the same way, we get

By continuous this process, we get all the r-Dowling numbers

As you can see from (39), the larger n, the more difficult it is to calculate by hand. Here we use Mathematica and Fortran language to find these values.

In Fig. 1, when

In Fig. 2, when

In Fig. 3, when

In Fig. 4, when

Next, we can get differential equation for degenerate r-Dowling polynomials as follows:

Theorem 3.11. For

Proof. By using Theorem 3.4, we observe

On the other hand, we have

From (42), we get

By (43), we obtain the desire result.

Now, we study the derivative of degenerate r-Dowling polynomials

Theorem 3.12. For

Proof. From (5) and Theorem 3.3, we observe that

By comparing the coefficients on both sides of (44), we attain the desired identity.

Theorem 3.13. For

Proof. From (5) and Theorem 3.3, we have

From (45), we observe that

By (46), we have

From (47), we obtain

By comparing the coefficients of both sides of (45), we have the desired identity.

If we put

Theorem 3.14. For

Proof. Let

Thus, by (49), we have

From (50), we attain the desired formula.

In this paper, we studied many interesting properties for the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind. Among these identity expressions, we obtained the generating function in Theorem 3.3, Dobinski-like formula in Theorem 3.4, recurrence relations in Theorem 3.6 and 3.8, differential equation in Theorem 3.11, the derivatives of them in Theorem 3.12 for r-Dowling polynomials of the second kind. In particular, we obtained some expressions for them that can be derived from repeated applications of certain operators to the exponential functions in Theorem 3.9, 3.10 and 3.14, and some identities involving integration in Theorem 3.13. Furthermore, we found that all exact values of all r-Dowling numbers of the second kind can be obtained using (28). As a follow-up study of this paper, we can explore truncated degenerate r-Dowling polynomials and degenerate r-Dowling polynomials arising from

Acknowledgement: The author would like to thank the referees for the detailed and valuable comments that helped improve the original manuscript in its present form. Also, the authors thank Jangjeon Institute for Mathematical Science for the support of this research.

Authors’ Contributions: HKK structured and wrote the whole paper. DSL performed computer simulations in the paper. All authors checked the results of the paper and completed the revision of the article.

Consent for Publication: The authors want to publish this paper in this journal.

Ethics Approval and Consent to Participate: The authors declare that there is no ethical problem in the production of this paper.

Funding Statement: This work was supported by the Basic Science Research Program, the National Research Foundation of Korea (NRF-2021R1F1A1050151).

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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