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 byand the Stirling numbers of the second kind are given bywhere and (see [4, 5]).
From (4) and (5), we can derive the following equations:(see [4–6]).
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 bewhere (see [10]). By (10), we know thatand thus, we obtain
In [11], the degenerate Bernoulli polynomials are defined aswhich 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 haveand so, we obtain
Note that by (3), we haveand 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 aswhere 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 thatand thus, we know that
Sinceby (29) and (30), we havewhere .
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 haveBy (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 , thenand 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),
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 haveand 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.