Journal of Mathematics
Volume 2020 (2020), Article ID 7893498, 9 pages
https://doi.org/10.1155/2020/7893498

Research Article

On Fully Degenerate Daehee Numbers and Polynomials of the Second Kind

Sang Jo Yun¹ and Jin-Woo Park²

¹Department of Mathematics, Dong-A University, Busan 604-714, Republic of Korea
²Department of Mathematics Education, Daegu University, Gyeongsan-si, Gyeongsangbuk-do 38453, Republic of Korea

Correspondence should be addressed to Jin-Woo Park; a0417001@knu.ac.kr

Received 31 May 2020; Accepted 20 July 2020; Published 2 September 2020

Academic Editor: Serkan Araci

Copyright © 2020 Sang Jo Yun and Jin-Woo Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In a study, Carlitz introduced the degenerate exponential function and applied that function to Bernoulli and Eulerian numbers and degenerate special functions have been studied by many researchers. In this paper, we define the fully degenerate Daehee polynomials of the second kind which are different from other degenerate Daehee polynomials and derive some new and interesting identities and properties of those polynomials.

1. Introduction

Let be a fixed prime number. Throughout this paper, , , and will denote the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively.

Let be a uniformly differentiable function on . Then, the -adic invariant integral on is defined as

(see [1–3]).

From (1), we have

In particular, if , then

The Stirling numbers of the first kind are defined by

and the Stirling numbers of the second kind are given by

where and (see [4, 5]).

From (4) and (5), we can derive the following equations:

(see [4–6]).

In addition,

[4, 5].

The Bernoulli polynomials of order are defined by the following generating function:

(see [7–9]).

Carlitz’s degenerate Bernoulli polynomials of order is defined by the generating function to be

where (see [10]). By (10), we know that

and thus, we obtain

In [11], the degenerate Bernoulli polynomials are defined as

which are different from Carlitz’s degenerate numbers and polynomials.

By (13), we know that

Note that by (3),

By (15) and (16), we know that

The higher-order Daehee polynomials are defined by the generating function to be

(see [12–15]).

In particular, if , then is called the Daehee polynomials.

By replacing as in (18), we have

and so, we obtain

Note that by (3), we have

and thus, we know that

Carlitz introduced the degenerate Bernoulli polynomials in [10] and the degeneration of special functions have been studied (see [6, 16–29]).

In particular, the degenerate Stirling numbers of the second kind with a generating function are defined as

where is a given nonnegative integer in [6, 10, 22, 30].

After introducing Daehee numbers and polynomials [31], it plays an important role of developing various generalized polynomials, and interesting properties are obtained (see [8, 15, 21, 22, 28, 30–35]).

In this paper, we define the new degenerate Daehee polynomials and numbers which are called the degenerate Daehee polynomials of the second kind and investigate identities and properties of new polynomials.

2. Fully Degenerate Daehee Polynomials of the Second Kind

Let us assume that . By (3), we have

By (24), we define the degenerate Daehee polynomials of the second kind by the generating function to be

In the special case, , and are called the degenerate Daehee numbers of the second kind.

Note that

and thus, we know that

From (7) and (24), we have

By (25) and (28), we have

Since

by (29) and (30), we have

where .

Thus, by (29) and (30), we have the following theorem which is Witt’s type formula about degenerate Daehee polynomials of the second kind.

Theorem 1. For each , we have

By replacing as in (25), we obtain the following:

On the other hand,

where is the degenerate Bernoulli polynomials of the second kind of order which are defined by the generating function to be

(see [20, 25]).

In particular, if , is called the degenerate Bernoulli polynomials of the second kind.

For positive integer with , if we put , then, by (2), we obtain

By (36), we have

Note that by (6),

By (24), (37), and (38), we have

Hence, by (33), (34), and (39), we obtain the following theorem which shows the relationship between degenerate Daehee polynomials of the second kind and degenerate Bernoulli polynomials of the second kind.

Theorem 2. For nonnegative integer and with , we have

By (25), we note that

By comparing the coefficients on both sides of (41), we obtain the following theorem.

Theorem 3. For nonnegative integer , we have

Note that if we put , then

and thus, by (3), we have

Moreover,

and, by (43), we obtain

From (43), (44), and (46), we obtain the following theorem which represents a recurrence relations between degenerate Daehee polynomials of the second kind and degenerate Daehee numbers of the second kind.

Theorem 4. For each nonnegative integer , we have

Moreover, for each positive integer ,

3. Higher-Order Degenerate Daehee Polynomials of the Second Kind

In this section, we consider the higher-order degenerate Daehee polynomials of the second kind given by the generating function as follows: for the given positive real number ,

In particular, if , are called the higher-order degenerate Daehee numbers of the second kind.

Note that

From (35), we note that

In addition, by replacing by in (49), we have

Hence, by (51)–(53), we obtain the following theorem.

Theorem 5. For , we have

Moreover,

In particular, if , then we know that Theorem 5 is a generalization of Theorem 1 and Theorem 2.

Note that for each ,

It is well known that for each ,

By (23), (7), and (57), we have

By (56) and (58), we obtain the following theorem.

Theorem 6. For each , we have

Note that by (3),

By (17) and (60), we have

Thus, by (60) and (61), we obtain the following theorem which shows that higher-order degenerate Daehee polynomials of the second kind are represented by linear combination of the higher-order Carlitz’s type degenerate Bernoulli polynomials.

Theorem 7. For each nonnegative integer and each integer ,

By (7) and (23), we have

and so by (22), (60), and (63), we obtain

By (60) and (64), we obtain the following theorem.

Theorem 8. For each nonnegative integer ,

Theorem 8 shows that higher-order degenerate Daehee polynomials are related closely to Daehee polynomials of order .

4. Conclusion

In the past two decades, the degenerations of special functions and their applications have been studied as a new area of mathematics. In this paper, we considered the degenerate Daehee numbers and polynomials by using -adic invariant integral on which are different from Kim’s degenerate Daehee polynomials. We derive some new and interesting properties of those polynomials.

Next, from the definition of the higher-order degenerate Daehee numbers and Daehee polynomials of the second kind, we found the relationship between the degenerate Bernoulli polynomials, the first and second Stirling numbers, the Bernoulli polynomials, degenerate Stirling numbers of the second kind, and those numbers and polynomials.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgments

This research was supported by the Daegu University Research Grant, 2019.

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