Type 2 degenerate modified poly-Bernoulli polynomials arising from the degenerate poly-exponential functions

1.   Introduction

The degenerate types of Bernoulli and Euler polynomials are initially introduced by Carlitz ^[3,4] with arithmetic and combinatorial results. Recently, these degenerate version of special functions and numbers have been intensively studied by many authors introducing various types of polynomials, functions and numbers (see ^{[1,5,6,7,8,10,11,12,15,17]} for which a rich variety of literature is available and references therein for further details). These special polynomials and numbers have been significant roles not only in combinatorial and arithmetic problems, but also various branches of problems arising in mathematics, theoretical physics, and engineering in order to new means of mathematical approaches.

In particular, a lot of research focusing on type 2 degenerate polynomials has been conducted: for example, the type 2 degenerate central Fubini polynomials ^[16], the type 2 degenerate poly-Bernoulli numbers and polynomials ^[11], the type 2 degenerate Euler and Bernoulli polynomials ^[6], the type 2 poly-Apostol–Bernoulli polynomials ^[9], the degenerate poly-type 2-Bernoulli Polynomials ^[1], the type 2 degenerate Euler and Bernoulli polynomials ^[6], a modification of the type 2 degenerate poly-Bernoulli polynomials ^[13], and the partially degenerate polyexponential-Bernoulli polynomials of the second kind ^[2].

Following by these studies, we introduce a new type of degenerate poly-Bernoulli polynomials and unipoly-Bernoulli polynomials called the type 2 degenerate modified poly-Bernoulli polynomials and the type 2 degenerate modified unipoly-Bernoulli polynomials attached to polynomials, respectively. We investigate their useful properties as well as their relations to express the proposed polynomials in terms of existing types or known functions and numbers. Further, we calculate some of the introduced polynomials and present their behaviors as well as their zeros for different variables in specific cases.

We first recall several definitions in order to introduce our new type of poly-Bernoulli polynomials.

The Bernoulli polynomials (see ^[10,11] for detail) are given by the generating function

and for case $x = 0$, $B_n: = B_n(0)$ are called the Bernoulli numbers.

The polyexponential functions are introduced in ^[8,11] for $k\in\mathbb{Z}$, which are defined by

For example, in case $k = 1$, it satisfies that ${Ei}_1(t) = e^t-1$, so that ${Ei}_1\left(\log(t+1)\right) = t$. For $k\leq0$, ${Ei}_k(t)$ are expressed as a product of polynomials with the exponential function such that

which are obtained from the recurrence relation $\frac{d}{dt} {Ei}_k(t) = \frac{1}{t} {Ei}_{k-1}(t)$ (see ^[11] for detail).

As a degenerate type of the polyexponential functions in (1.2), the author ^[11] considered the modified degenerate polyexponential functions for $k\in \mathbb{N}$ given by

where $(t)_{n, \lambda}$ is the $\lambda$-falling factorial sequence ^[11] defined by $(t)_{n, \lambda} = t(t-\lambda)(t-2\lambda)\cdots (t-(n-1)\lambda)$ for $n\geq 1$ and $(t)_{0, \lambda} = 1$. It is well known that ${Ei}_{1, \lambda}(t) = e_\lambda(t)-1$ and ${Ei}_{1, \lambda}\left(\log_\lambda (1+t)\right) = t$.

For $\lambda \in\mathbb{R}\backslash\{0\}$, the author in ^[3,4] introduced the degenerate Bernoulli polynomials and the degenerate Bernoulli numbers, which are respectively defined by

where the degenerate exponential functions are given by

As a variant of degenerate Bernoulli polynomials, the author in ^[1] considered the degenerate poly-type 2-Bernoulli polynomials $B_{n, \lambda}^{(k)}$, which are defined by the generating function

where ${Li}_k(t)$ for $k\in \mathbb{Z}$ are the polylogarithm functions ^[5] defined by

For the case $k = 1$ of $B_{n, \lambda}^{(k)}$, $B_{n, \lambda}^{(1)}(x): = \beta_{n, \lambda}(x)$ are called the degenerate type 2-Bernoulli polynomials, which satisfy

since the identity holds ${Li}_1(1-e^{-t}) = t$ for $k = 1$.

In ^[10], another variant of the degenerate Bernoulli polynomials, called the degenerate Bernoulli polynomials of the second kind, is introduced by modifying the generating function such as

Here the degenerate logarithm function $\log_\lambda (t) = \frac{1}{\lambda}(t^\lambda-1)$ is the compositional inverse of $e_\lambda (t)$, i.e., $\log_\lambda \left(e_\lambda (t)\right) = e_\lambda\left(\log_\lambda (t)\right) = t$.

Further, applying the polyexponential function (1.2) to the degenerate Bernoulli polynomials, the type 2 degenerate poly-Bernoulli polynomials ^[11] are introduced by the following formula

in which, for case $x = 0$, $B_{n, \lambda}^{(k)}: = B_{n, \lambda}^{(k)}(0)$ are called the type 2 degenerate poly-Bernoulli numbers.

As a different variation of (1.8) using the polyexponential function, the type 2 degenerate poly-Bernoulli polynomials ^[14] of the second kind are introduced by the generating function

In further research, the type 2 degenerate Euler polynomials are introduced by the author in ^[6] based on the following generating function

and $\mathcal{E}_{n, \lambda}: = \mathcal{E}_{n, \lambda}(0)$ are called the type 2 degenerate Euler numbers when $x = 0$.

2.   Type 2 degenerate modified poly-Bernoulli polynomials

In this section, motivated by the previous mentioned works about variant types of degenerate poly-Bernoulli polynomials, we introduce a new type of degenerate poly-Bernoulli polynomials and discuss their properties.

Definition 2.1. Let us define the type 2 degenerate modified poly-Bernoulli polynomials $\mathcal{B}^{[k]}_{n, \lambda}(x)$ for $k\in\mathbb{Z}$ by the generating function given by

where ${{Ei}}_{k, \lambda}$ are the degenerate modified polyexponential function given in $(1.4)$. Especially, when $x = 0$, $\mathcal{B}^{[k]}_{n, \lambda}: = \mathcal{B}^{[k]}_{n, \lambda}(0)$ are called the type 2 degenerate modified poly-Bernoulli numbers.

Remark 2.1. Since ${Ei}_{1, \lambda}\left(\log_\lambda (1+t)\right) = t$ for $k = 1$, one can see that $\mathcal{B}^{[1]}_{n, \lambda}(x)$ satisfy

which provides the same generating function listed in $(1.7)$, so that $\mathcal{B}^{[k]}_{n, \lambda}(x)$ represent the degenerate type 2-Bernoulli polynomials $\beta_{n, \lambda}(x)$ for $k = 1$. Thus, the type 2 degenerate modified poly-Bernoulli polynomials $\mathcal{B}^{[k]}_{n, \lambda}(x)$ is a generalization of the degenerate type 2-Bernoulli polynomials $\beta_{n, \lambda}(x)$, while the type 2 degenerate poly-Bernoulli polynomials $(1.10)$ of the second kind is a generalization of the degenerate Bernoulli polynomials of the second kind.

First, we can express the type 2 degenerate modified poly-Bernoulli polynomials in terms of the degenerate Stirling numbers of the first kind and the degenerate type 2-Bernoulli polynomials (1.7).

Theorem 2.1. For $n\geq0$ and $\lambda \in \mathbb{R}\backslash\{0\}$, the following identity holds:

where $S_{1, \lambda}(n, m)$ are the degenerate Stirling numbers of the first kind defined by

Proof. We first note (see ^[11] for detail) that the degenerate polyexponential functions ${Ei}_{k, \lambda}(t)$ satisfy that

From (1.7), (2.1) and Definition 2.1, we can see that

By comparing the coefficients on the both sides of (2.2), one can conclude the desired result.

Remark 2.2. (1) One can see that the type 2 degenerate poly-Bernoulli polynomials $(1.10)$ of the second kind are expressed in terms of the degenerate Stirling numbers of the first kind and the degenerate Bernoulli polynomials of the second kind with the same combinatorial coefficients (see [14,Theorem 2.1]) as a similar expression.

(2) From $\frac{d}{dt} {Ei}_{k, \lambda} \left(\log_{\lambda}(1+t)\right) = \frac{(1+t)^{\lambda-1}}{\log_{\lambda}(1+t)} {Ei}_{k-1, \lambda} \left(\log_{\lambda}(1+t)\right)$ for $k\geq 2$ (see ^[11] for detail), ${Ei}_{k, \lambda}\left(\log_\lambda(1+t)\right)$ can be expressed as the multiple integrals of ${Ei}_{1, \lambda}\left(\log_\lambda(1+t)\right)$ as follows:

in which $k-1$ times of integrals are performed by the product with $\frac{(1+t)^{\lambda-1}}{\log_{\lambda}(1+t)}$ each time.

Next, using the property of Remark 2.1, we have the following result.

Theorem 2.2. For $n\geq 0, \; k\in \mathbb{Z},$ and $\lambda \in \mathbb{R}\backslash\{0\}$, $\mathcal{B}^{[k]}_{n, \lambda}(x)$ satisfy the following identity:

Proof. By making use of Definition 2.1, (1.7), (1.8) and Remark 2.1, we can verify that

Hence, the conclusion is established by comparing coefficients on both sides.

We next give the expression for $\mathcal{B}^{[k]}_{n, \lambda}(x)$ as a sum of the products of the type 2 degenerate modified poly-Bernoulli numbers and $\lambda$-falling factorial sequence.

Theorem 2.3. Let $k$ be any integer. Then the following identity holds true for all $n\geq 0$,

Proof. By considering $\mathcal{B}^{[k]}_{n, \lambda}(x)$ to be the product of $\frac{ {{Ei}}_{k, \lambda}(\log_\lambda (1+t))}{e_{\lambda}(t)-e_\lambda^{-1}(t)}$ and ${e_\lambda^x (t)}$ and using the identity of ${e_\lambda^x (t)} = \sum_{m = 0}^{\infty} (x)_{m, \lambda}\frac{t^m}{m!}$, the desired result for $\mathcal{B}^{[k]}_{n, \lambda}(x)$ is obtained by the binomial convolution of the sequences $\left\{ \mathcal{B}^{[k]}_{n, \lambda} \right\}_{n = 0}^\infty$ and $\left\{ (x)_{n, \lambda} \right\}_{n = 0}^\infty$.

Let us present the recurrence relation for $\mathcal{B}^{[k]}_{n, \lambda}(x)$ from the derivations of the expressions for ${{Ei}}_{k, \lambda}\left(\log_\lambda (1+t)\right)$.

Theorem 2.4. For $n\geq 1$ and $k\in\mathbb{Z}$, the following relationholds:

Proof. We first note from ^[10] that

On the other hand, in the virtue of the result of Theorem 2.3, it follows that

Therefore, we can show the desired relation by comparing coefficients of (2.7) and (2.8).

Also, $\mathcal{B}^{[k]}_{n, \lambda}(x)$ can be expressed in terms of the type 2 degenerate poly-Bernoulli numbers and the type 2 degenerate Euler polynomials.

Theorem 2.5. For $n\geq 0$ and $k\in\mathbb{Z}$, $\mathcal{B}^{[k]}_{n, \lambda}(x)$ can be expressed as the product of $B_{n, \lambda}^{(k)}$ and $\mathcal{E}_{n, \lambda}(x)$ such as

Proof. With the help of (1.9) and (1.11), one can have

Thus, the previous identity gives the identity (2.9) by the comparing coefficients.

We close this section with an expression for $\mathcal{B}^{[k]}_{n, \lambda}(x)$ in terms of ${\mathcal{B}^{[1]}_{n, \lambda}}$ and $S_{1, \lambda}(n, m)$.

Theorem 2.6. For $n\geq 0$ and $k\in\mathbb{Z}$, the following identity holds

Proof. Note that from (2.1)

From the previous note and Remark 2.1, the followings are established:

Therefore, the assertion is obtained by comparing coefficients.

3.   Type 2 degenerate modified unipoly-Bernoulli polynomials attached to polynomials

In this section, we consider a new type of the degenerate unipoly-Bernoulli polynomials attached to $p$, where $p$ is any arithmetic function defined on $\mathbb{N}$. The unipoly function attached to $p$ is recently introduced in ^[8], which is defined by

Applying the degenerate type of unipoly function attached to $p$ in the degenerate poly-Bernoulli polynomials of type 2, we introduce a new type of degenerate unipoly polynomials.

Definition 3.1. Let $k$ be any integer. Then we define the type 2 degenerate modified unipoly-Bernoulli polynomials attached to polynomials $p$, which are given by

where $u_{k, \lambda}(x|p)$ is the degenerate unipoly function ^[8,11] attached to any arithmetic function $p(n)$ defined by

In particular, when $x = 0$, ${\mathcal{PB}^{[k]}_{n, \lambda, p} } = {\mathcal{PB}^{[k]}_{n, \lambda, p} } (0)$ are called the type 2 degenerate modified unipoly-Bernoulli numbers.

We note that if the attached arithmetic function satisfies $p(n) = \frac{1}{\Gamma(n)}, \; n\in\mathbb{N}$, it is seen that

Also, one can see that ${\mathcal{PB}}^{[1]}_{n, \lambda, \frac{1}{\Gamma}}(x) = \beta_{n, \lambda}(x)$ when $k = 1$ from (1.7) and the identity holds

For example, using the scientific calculator, we can see that the first five polynomials ${\mathcal{PB}^{[1]}_{k, \lambda, \frac{1}{\Gamma}} } (x), \; k = 0, \cdots, 4$, are estimated as

First, we show that ${\mathcal{PB}^{[k]}_{n, \lambda, p}} (x)$ can be expressed in terms of ${\mathcal{B}}_{n, \lambda}^{[1]}(x)$ and the degenerate Stirling numbers of the first kind.

Theorem 3.1. For $k\in\mathbb{Z}, n\geq0, \lambda \in \mathbb{R}\backslash\{0\}$, the following identity is established:

Proof. It is well known that

and

where

which can be obtained from $\left(\log_{\lambda}(1+t)\right)^{n+1}/{(n+1)!} = \sum_{m = n+1}^{\infty}S_{1, \lambda}(m, n+1)\frac{t^m}{m!}.$ Since the product of the left hand sides of Eqs (3.2) and (3.3) provides the generating function in Definition 3.1, we obtain the desired identity by the binomial convolution of $\left\{ \mathcal{B}^{[1]}_{n, \lambda}(x)\right\}_{n = 0}^{\infty}$ and $\left\{C_{n, k, \lambda}\right\}_{n = 0}^{\infty}$.

We next show that ${\mathcal{PB}^{[k]}_{n, \lambda, p}} (x)$ is expressed as a sum of the products of the type 2 degenerate modified unipoly-Bernoulli numbers and $\lambda$-falling factorial sequence for $x$.

Theorem 3.2. Let $k\in\mathbb{Z}, n\geq0, \lambda \in \mathbb{R}\backslash\{0\}$. Then we have the identity:

Proof. By making use of the identities

one can obtain the result from the binomial convolution of two identities.

Finally, we present the type 2 degenerate unipoly-Bernoulli numbers for $k = 1$ can be considered to the sum of the type 2 degenerate Euler polynomials (1.11) and the degenerate Bernoulli numbers (1.5).

Theorem 3.3. Let $p(n) = \frac{1}{\Gamma(n)}$ for $n\geq0, \lambda \in \mathbb{R}\backslash\{0\}$, we have

Proof. Recalling from Definition 3.1 for $k = 1$ that

in which if $x = 0$, the right hand side of the previous equation provides that

with the help of the definitions (1.5) and (1.11). Hence, the identity follows by comparing coefficients.

4.   Illustrative examples

In this section, we present several examples and some graphical representations of the type 2 degenerate modified poly-Bernoulli polynomials and the type 2 degenerate modified unipoly-Bernoulli polynomials attached to polynomials presented in the previous sections.

We first note that by using the recurrence relation $\frac{d}{dt} {Ei}_{k, \lambda}(t) = \frac{1}{t} {Ei}_{k-1, \lambda}(t)$ with the property of $\frac{d}{dt}e_{\lambda}(t) = \frac{1}{1+\lambda t}e_{\lambda}(t)$, ${Ei}_{k, \lambda}(t)$ for $k\leq 1$ can be calculated as follows:

Remark 4.1. From $(1.3)$ and $(4.1)$, we see that the polyexponential functions ${Ei}_{k}(t)$ are considered as the limit of the degenerate polyexponential functions ${Ei}_{k, \lambda}(t)$ as $\lambda$ approaches to $0$, i.e., $\lim_{\lambda\to 0} {Ei}_{k, \lambda}(t) = {Ei}_{k}(t).$

In order to see some of the type 2 degenerate modified poly-Bernoulli polynomials, we calculate the first six polynomials of $\mathcal{B}^{[1]}_{n, \lambda}(x)$ for $k = 1$, which are listed below:

Further, following the presentation in ^[18], we investigate the zeros of $\mathcal{B}^{[1]}_{n, \lambda}(x), n = 1, 2, \cdots, 8$ for $\lambda = \frac{1}{4}$ in Table 1 and plot the zeros of ${\mathcal{B}^{[1]}_{n, \lambda} }(x) = 0$ for $n = 1, \cdots, 15$ when $\lambda = 1/4$ in Figures 1 and 2.

Also, we plot the behaviour of the type 2 degenerate modified unipoly-Bernoulli polynomials, ${\mathcal{PB}^{[1]}_{6, \lambda, \frac{1}{\Gamma}}} (x)$ and ${\mathcal{PB}^{[1]}_{7, \lambda, \frac{1}{\Gamma}}} (x)$ attached to polynomials $p = \frac{1}{\Gamma}$ in Figure 3 for various values of $\lambda = 30, 25, \cdots, 1/20$. The figures show the degenerate polynomials ${\mathcal{PB}^{[1]}_{6, \lambda, \frac{1}{\Gamma}}} (x)$ and ${\mathcal{PB}^{[1]}_{7, \lambda, \frac{1}{\Gamma}}} (x)$ approach to the type 2 Bernoulli polynomials $B^*_6(x)$ and $B^*_7(x)$, respectively, as $\lambda$ is getting small, where $B^*_n(x)$ are defined by the generating function ^[6]

5.   Conclusions

Following recent research about type 2 degenerate polynomials, we studied the new types of degenerate poly-Bernoulli polynomials and degenerate unipoly-Bernoulli polynomials, which are obtained by modifying generating functions based on degenerate exponential functions and degenerate logarithm functions. We present the general properties of the polynomials including recursion relation and related identities in terms of well known special functions and numbers. Furthermore, in order to see their zeros and behaviors, we calculate some of the proposed polynomials for specific cases and display their zeros and polynomials for different variables. In general, the relation among special polynomials provides not only an important research topic but also useful identities in physics, science and engineering as well as in mathematics. Thus, it would be interested to find the relation among the variants of some special polynomials and their applications, which are one of our future work.

Acknowledgments

This work was supported by the Dongguk University Research Fund of 2020 and the corresponding author (Kwon) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. 2020R1F1A1A0107156412).

The authors would like to thank the referees for their careful reading and helpful comments.

Conflict of interest

The authors declare no conflict of interest.