Abstract
In this paper, we introduce degenerate multi-poly-Bernoulli polynomials and derive some identities of these polynomials. We give some relationship between degenerate multi-poly-Bernoulli polynomials degenerate Whitney numbers and Stirling numbers of the first kind. Moreover, we define degenerate multi-poly-Bernoulli polynomials of complex variables, and then, we derive several properties and relations.
1. Introduction
For any (or ), degenerate version of the exponential function is defined as follows (see [1–15]) where and for , (cf. [1–15]). It follows from (1) is . Note that
Carlitz [1] introduced the degenerate Bernoulli polynomials as follows:
Upon setting , are called the degenerate Bernoulli numbers.
Note that where are the familiar Bernoulli polynomials (cf. [1, 3, 4, 6, 8, 11, 12, 14, 16–22])
For , the polyexponential function is defined by (see [21]) Setting in (5), we have
The degenerate modified polyexponential function [12] is defined, for and , by Note that
Let and . The degenerate poly-Bernoulli polynomials, cf. [12], are defined by where are called the degenerate version of the logarithm function (cf. [8, 12]), which is also the inverse function of the degenerate exponential function as shown below (cf. [8]) Letting in (7), are called the type 2 degenerate poly-Bernoulli numbers.
The degenerate Stirling numbers of the first kind (cf. [8, 13]) and second kind (cf. [4–6, 9, 17]) are defined, respectively, by and (cf. [1–27]) Note that in (10) and (1.8), we have (cf. [8, 13]) and (cf. [4–6, 9, 17, 24]) where and are called the Stirling numbers of the first kind and second kind.
The following paper is as follows. In Section 2, we define the degenerate multi-poly-Bernoulli polynomials and numbers by using the degenerate multiple polyexponential functions and derive some properties and relations of these polynomials. In Section 3, we consider the degenerate multi-poly-Bernoulli polynomials of a complex variable and then we derive several properties and relations. Also, we examine the results derived in this study [28, 29].
2. Degenerate Multi-Poly-Bernoulli Polynomials and Numbers
Let . The degenerate multiple polyexponential function is defined (cf. [15]) by where the sum is over all integers satisfying . Utilizing this function, Kim et al. [15] introduced and studied the degenerate multi-poly-Genocchi polynomials given by Inspired by the definition of degenerate multi-poly-Genocchi polynomials, using the degenerate multiple polyexponential function (14), we give the following definition.
Definition 1. Let and , we consider the degenerate multi-poly-Bernoulli polynomials are given by
Upon setting in (16), the degenerate multi-poly-Bernoulli polynomials reduce to the corresponding numbers, namely, the type 2 degenerate multi-poly-Bernoulli numbers .
Remark 2. As the degenerate multi-poly-Bernoulli polynomials reduce to the multi-poly-Bernoulli polynomials given by
Remark 3. Upon setting in (16), the degenerate multi-poly-Bernoulli polynomials reduce to the degenerate poly-Bernoulli polynomials in (7).
Before going to investigate the properties of the degenerate multi-poly-Bernoulli polynomials, we first give the following result.
Proposition 4 (Derivative Property). For and , we have
Proof. By (14), we see that
Theorem 5. The following relationship holds for .
Proof. Recall Definition 1 that which gives the asserted result (20).
The degenerate Bernoulli polynomials of order are given by the following series expansion: (cf. [3, 6, 8, 17]).
We provide the following theorem.
Theorem 6. For . Then
Proof. Recall from Definition 1 and (10) that which means the claimed result (23).
Theorem 7. The following formula is valid for and .
Proof. In view of Definition 1, we see that which implies the desired result (25).
Theorem 8. The following relation is valid for and .
Proof. To investigate the derivative property of that which provides the asserted result (27).
We here give a relation including the degenerate multi-poly-Bernoulli polynomials with numbers and the degenerate Stirling numbers of the second kind.
Theorem 9. The following correlation is valid for and .
Proof. By means of Definition 1, we attain that where the notation is falling factorial that is defined by and for , (cf. [1, 2, 5–14, 21, 23, 24]). So, the proof is completed.
Kim [5] introduced the degenerate Whitney numbers are given by Kim also provided several correlations including the degenerate Stirling numbers of the second kind and the degenerate Whitney numbers (see [5]).
We now give a correlation as follows.
Theorem 10. For and , we have
Proof. Using (31) and Definition 1, we acquire that which implies the asserted result (32).
3. Degenerate Multi-Poly-Bernoulli Polynomials of Complex Variable
In [25], Kim et al. defined the degenerate sine and cosine functions by where . Note that and . From (34), it is readily that
By these functions in (34), the degenerate sine-polynomials and degenerate cosine-polynomials are introduced (cf. [25]) by Several properties of these polynomials in (36) and (37) are studied and investigated in [25]. Also, by means of these functions, Kim et al. [25] introduced the degenerate Euler and Bernoulli polynomials of complex variable and investigate some of their properties. Motivated and inspired by these considerations above, we define type 2 degenerate multi-poly-Bernoulli polynomials of complex variable as follows.
Definition 11. Let . We define a new form of the degenerate multi-poly-Bernoulli polynomials of complex variable by the following generating function:
By (34) and (38), we observe that and
In view of (39) and (40), we consider the degenerate multi-poly-sine-Bernoulli polynomials with two parameters and the degenerate multi-poly-cosine-Bernoulli polynomials with two parameters as follows:
Note that which are multi-poly-sine-Bernoulli polynomials and multi-poly-cosine-Bernoulli polynomials with two parameters.
We now give the two summation formulae by the following theorem.
Theorem 12. For and , we have
Proof. The proofs of this theorem can be done by using the same proof methods used in Theorems 5 and 7. So, we omit the proofs.
We here provide the two derivative formulae by the following theorem.
Theorem 13. For and , we have
Proof. The proofs of this theorem can be done by using the same proof methods used in Theorem 8. So, we omit the proofs.
We give the following theorem.
Theorem 14. For and , we have
Proof. From (36), (37), and (38), we get which complete the proof of the theorem.
We note that (cf. [25])
We give the following theorem.
Theorem 15. For and , we have where the notation is Gauss’ notation and represents the maximum integer which does not exceed a number in the square bracket.
Proof. By (41)–(51), we observe that So, the proof is completed.
We give the following proposition.
Proposition 16. The following relations hold for and .
Proof. The proofs of this proposition can be done by utilizing the same proof methods used in Theorem 7. So, we omit the proofs.
Upon setting in (41) and (42), we consider the degenerate multi-poly-sine-Bernoulli polynomials and the degenerate multi-poly-cosine-Bernoulli polynomials as follows We now provide the following theorem.
Theorem 17. For and , we have
Proof. The proofs of this theorem can be done by utilizing the same proof methods in Theorem 9.
Let be any fixed real (or complex) number. The Bernoulli polynomials of order is defined by (cf. [25]) When , the Bernoulli polynomials of order reduce to the Bernoulli numbers of order , denoted by .We give the following relation.
Theorem 18. For and , we have
Proof. By (55) and (56), we acquire Thus, (59) is proved. We prove (60) in the same way.
Here is a special case of Theorem 18.
Corollary 19. For and , we have which is a relation including the degenerate multi-poly-Bernoulli polynomials.
4. Conclusions
In this paper, we defined the degenerate multi-poly-Bernoulli polynomials by employing the degenerate multiple polyexponential functions. We have established some identities and relations between degenerate Whitney numbers and degenerate Stirling numbers of the first kind. Also, we have established addition formulas and derivative formulas of degenerate multi-poly-Bernoulli polynomials. In the last section, we have defined degenerate multi-poly-Bernoulli polynomials of complex variables and then we have derived several properties and relations.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.