A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

Waseem A. Khan; Abdulghani Muhyi; Rifaqat Ali; Khaled Ahmad Hassan Alzobydi; Manoj Singh; Praveen Agarwal; Waseem A. Khan; Abdulghani Muhyi; Rifaqat Ali; Khaled Ahmad Hassan Alzobydi; Manoj Singh; Praveen Agarwal

doi:10.3934/math.2021731

AIMS Mathematics

2021, Volume 6, Issue 11: 12680-12697. doi: 10.3934/math.2021731

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Research article Special Issues

A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

1.
Department of Mathematics and Natural Sciences, Prince Mohammad Bin Fahd University, P.O Box 1664, Al Khobar 31952, Saudi Arabia
2.
Department of Mathematics, Hajjah University, Hajjah, Yemen
3.
Department of Mathematics, College of Science and Arts, Muhayil, King Khalid University, P.O Box 9004, Postal Code:61413. Abha, Saudi Arabia
4.
Department of Mathematics, College of Science, Jazan University, Jazan, Saudi Arabia
5.
Department of Mathematics, Anand International College of Engineering, Jaipur 303012, India
6.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman AE 346, United Arab Emirates
7.
International Center for Basic and Applied Sciences, Jaipur 302029, India

Received: 19 July 2021 Accepted: 18 August 2021 Published: 06 September 2021
MSC : 05A19, 11B73, 11B83

The main object of this article is to present type 2 degenerate poly-Bernoulli polynomials of the second kind and numbers by arising from modified degenerate polyexponential function and investigate some properties of them. Thereafter, we treat the type 2 degenerate unipoly-Bernoulli polynomials of the second kind via modified degenerate polyexponential function and derive several properties of these polynomials. Furthermore, some new identities and explicit expressions for degenerate unipoly polynomials related to special numbers and polynomials are obtained. In addition, certain related beautiful zeros and graphical representations are displayed with the help of Mathematica.

Keywords:

Citation: Waseem A. Khan, Abdulghani Muhyi, Rifaqat Ali, Khaled Ahmad Hassan Alzobydi, Manoj Singh, Praveen Agarwal. A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties[J]. AIMS Mathematics, 2021, 6(11): 12680-12697. doi: 10.3934/math.2021731

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Abstract

1. Introduction

In ^[1,2], Carlitz initiated study of the degenerate Bernoulli and Euler polynomials and obtained some arithmetic and combinatorial results on them. In recent years, many mathematicians have drawn their attention to various degenerate versions of some old and new polynomials and numbers, namely some degenerate versions of Bernoulli numbers and polynomials of the second kind, Changhee numbers of the second kind, Daehee numbers of the second kind, Bernstein polynomials, central Bell numbers and polynomials, central factorial numbers of the second kind, Cauchy numbers, Eulerian numbers and polynomials, Fubini polynomials, Stirling numbers of the first kind, Stirling polynomials of the second kind, central complete Bell polynomials, Bell numbers and polynomials, type 2 Bernoulli numbers and polynomials, type 2 Bernoulli polynomials of the second kind, poly-Bernoulli numbers and polynomials, poly-Cauchy polynomials, and of Frobenius-Euler polynomials, to name a few ^{[3,14,16,17,18]} and the references therein. They have studied those polynomials and numbers with their interest not only in combinatorial and arithmetic properties but also in differential equations and certain symmetric identities ^[4,5] and references therein, and found many interesting results related to them ^{[12,19,20,21,22,23,24,25,26,27,28]}. It is remarkable that studying degenerate versions is not only limited to polynomials but also extended to transcendental functions.

The Bernoulli polynomials of the second are defined by as follows (see ^[9,13])

$\begin{align} \frac{z}{\log(1+z)}(1+z)^x = \sum\limits_{q = 0}^{\infty}b_q(x)\frac{z^q}{q!}. \end{align}$

(1.1)

When $x = 0$ , $b_q(0) = b_q$ are called the Bernoulli numbers of the second kind.

The degenerate exponential function $e_{\lambda}^{x}(z)$ is defined by (see ^{[6,7,8,9,10,11,12,13,14,15,16,17,18,19]})

$\begin{align} e_{\lambda}^{x}(z) = (1+\lambda z)^{\frac{x}{\lambda}}, \; \; e_{\lambda}(z) = (1+\lambda z)^{\frac{1}{\lambda}}, \; \lambda\in \mathbb{C}\backslash \{0\}. \end{align}$

(1.2)

We note that

$\begin{align} e_{\lambda}^{x}(z) = \sum\limits_{q = 0}^{\infty}(x)_{q, \lambda}\frac{z^q}{q!}, {\rm{(see \; [4, 21])}}, \end{align}$

(1.3)

where $(x)_{q, \lambda} = x(x-\lambda)\cdots(x-(q-1)\lambda), (q\geq1)$ , $(x)_{0, \lambda} = 1$ .

Note that

$\lim\limits_{\lambda\rightarrow 0}e_{\lambda}^{x}(z) = \sum\limits_{q = 0}^{\infty}x^q\frac{z^q}{q!} = e^{xz}.$

The degenerate Bernoulli polynomials which are defined by Carlitz's as follows (see ^[1,2])

$\begin{align} \frac{z}{e_{\lambda}(z)-1}e_{\lambda}^{x}(z) = \frac{z}{(1+\lambda z)^{\frac{1}{\lambda}}-1}(1+\lambda z)^{\frac{x}{\lambda}} = \sum\limits_{q = 0}^{\infty}\beta_{q}(x;\lambda)\frac{z^q}{q!}. \end{align}$

(1.4)

At the point $x = 0$ , $\beta_{q}(\lambda) = \beta_{q}(0;\lambda)$ are called the degenerate Bernoulli numbers.

Note that

$\lim\limits_{\lambda \longrightarrow 0}\beta_{q}(x;\lambda) = B_{q}(x).$

The polylogarithm function is defined by

$\begin{align} {\rm Li}_{k}(x) = \sum\limits_{q = 1}^{\infty}\frac{x^q}{q^k}\; \; \; (k\in\mathbb Z, \; \mid x\mid < 1), {\rm{(see \; [7])}}. \end{align}$

(1.5)

Note that

$\begin{align} {\rm Li}_{1}(x) = \sum\limits_{q = 1}^{\infty}\frac{x^q}{q} = -\log(1-x). \end{align}$

(1.6)

The poly-Bernoulli polynomials of the second are defined by (see ^[13])

$\begin{align} \frac{{\rm Li}_{k}(1-e^{-z})}{\log(1+z)}(1+z)^{x} = \sum\limits_{q = 0}^{\infty}b_{q}^{(k)}(x)\frac{z^q}{q!}. \end{align}$

(1.7)

In the case when $x = 0$ , $b_{q}^{(k)} = b_{q}^{(k)}(0)$ are called the poly-Bernoulli numbers of the second kind.

The modified degenerate polyexponential function is defined by (see ^[14])

$\begin{align} {\rm Ei}_{k, \lambda}(x) = \sum\limits_{q = 1}^{\infty}\frac{(1)_{q, \lambda}}{(q-1)!q^k}x^q. \end{align}$

(1.8)

It is noteworthy to mention that

${\rm Ei}_{1, \lambda}(x) = \sum\limits_{q = 1}^{\infty}\frac{(1)_{q, \lambda}}{q!}x^q = e_{\lambda}(x)-1.$

The degenerate poly-Genocchi polynomials which are defined by Kim et al. as follows (see ^[14])

$\begin{align} \frac{2{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{e_{\lambda}(z)+1}e_{\lambda}^{x}(z) = \sum\limits_{q = 0}^{\infty}G_{q, \lambda}^{(k)}(x)\frac{z^q}{q!}\; \; \; (k\in\mathbb Z). \end{align}$

(1.9)

When $x = 0$ , $G_{q, \lambda}^{(k)} = G_{q, \lambda}^{(k)}(0)$ are called the degenerate poly-Genocchi numbers.

For $\lambda\in\mathbb R$ , Kim-Kim defined the degenerate version of the logarithm function, denoted by $\log_{\lambda}(1+t)$ as follows (see ^[11])

$\begin{align} \log_{\lambda}(1+z) = \sum\limits_{q = 1}^{\infty}\lambda^{\lambda-1}(1)_{q, 1/\lambda}\frac{z^q}{q!}, \end{align}$

(1.10)

being the inverse of the degenerate version of the exponential function $e_{\lambda}(z)$ as has been shown below

$e_{\lambda}(\log_{\lambda}(z)) = \log_{\lambda}(e_{\lambda}(z)) = z.$

It is noteworthy to mention that

$\lim\limits_{\lambda\rightarrow 0}\log_{\lambda}(1+z) = \sum\limits_{q = 1}^{\infty}(-1)^{q-1}\frac{z^q}{q!} = \log(1+z).$

The degenerate Daehee polynomials are defined by (see ^[15])

$\begin{align} \frac{\log_{\lambda}(1+z)}{z}(1+z)^{x} = \sum\limits_{q = 0}^{\infty}D_{q, \lambda}(x)\frac{z^q}{q!}. \end{align}$

(1.11)

In the case when $x = 0$ , $D_{q, \lambda} = D_{q, \lambda}(0)$ denotes the degenerate Daehee numbers.

The degenerate Bernoulli polynomials of the second kind which are defined by Kim et al. as follows (see ^[9])

$\begin{align} \frac{z}{\log_{\lambda}(1+z)}(1+z)^{x} = \sum\limits_{q = 0}^{\infty}b_{q, \lambda}(x)\frac{z^q}{q!}. \end{align}$

(1.12)

When $x = 0$ , $b_{q, \lambda} = b_{q, \lambda}(0)$ are called the degenerate Bernoulli numbers of the second kind.

Note here that $\lim_{\lambda \rightarrow 0}b_{q, \lambda}(x) = b_q(x), (q\geq 0)$ .

The degenerate Stirling numbers of the first kind are defined by

$\begin{align} \frac{1}{k!}(\log_{\lambda}(1+z))^k = \sum\limits_{q = k}^{\infty}S_{1, \lambda}(q, k)\frac{z^q}{q!}\; \; (k\geq 0), \rm{(see \; [11, 12])}. \end{align}$

(1.13)

It is noticed that

$\lim\limits_{\lambda\rightarrow 0}S_{1, \lambda}(q, k) = S_1(q, k),$

are the Stirling numbers of the first kind presented by

$\frac{1}{k!}(\log(1+z))^k = \sum\limits_{q = k}^{\infty}S_{1}(q, k)\frac{z^q}{q!}\; \; (k\geq 0), {\rm{(see \; [7, 17])}}.$

The degenerate Stirling numbers of the second kind are defined by (see ^[8])

$\begin{align} \frac{1}{k!}(e_{\lambda}(z)-1)^k = \sum\limits_{q = k}^{\infty}S_{2, \lambda}(q, k)\frac{z^q}{q!}\; \; (k\geq 0). \end{align}$

(1.14)

It is clear that

$\lim\limits_{\lambda\rightarrow 0}S_{2, \lambda}(q, k) = S_2(q, k),$

are the Stirling numbers of the second kind specified by

$\frac{1}{k!}(e^z-1)^k = \sum\limits_{q = k}^{\infty}S_{2}(q, k)\frac{z^q}{q!}\; \; (k\geq 0), {\rm{(see \; [1-28])}}.$

Motivated by the works of Kim et al. ^[11,14], in this paper, we study the type 2 degenerate poly-Bernoulli polynomials of the second kind arising from modified degenerate polyexponential function and obtain some related identities and explicit expressions. Also, we establish the type 2 degenerate unipoly-Bernoulli polynomials of the second kind attached to an arithmetic function by using modified degenerate polyexponential function and discuss some properties of them.

2. Type 2 degenerate poly-Bernoulli polynomials of the second kind

Here, the type 2 degenerate poly-Bernoulli polynomials of the second kind are defined by using the modified degenerate polyexponential function which is called the degenerate poly-Bernoulli polynomials of the second kind as

$\begin{align} \frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{\log_{\lambda}(1+z)}(1+z)^{x} = \sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}(x)\frac{z^j}{j!}, (k\in\mathbb Z). \end{align}$

(2.1)

When $x = 0$ , $Pb_{j, \lambda}^{(k)} = Pb_{j, \lambda}^{(k)}(0)$ are called the type 2 degenerate poly-Bernoulli numbers of the second kind.

Note that

$\lim\limits_{\lambda\rightarrow 0}\frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{\log_{\lambda}(1+z)}(1+z)^{x} = \sum\limits_{j = 0}^{\infty}\lim\limits_{\lambda\rightarrow 0}Pb_{j, \lambda}^{(k)}(x)\frac{z^j}{j!}$

$\begin{align} = \frac{{\rm Ei}_{k}\left(\log(1+z)\right)}{\log(1+z)}(1+z)^{x} = \sum\limits_{j = 0}^{\infty}Pb_{j}^{(k)}(x)\frac{z^j}{j!}, (k\in\mathbb Z), \end{align}$

(2.2)

where $Pb_{j}^{(k)}(x)$ are called the type 2 poly-Bernoulli polynomials of the second kind (see ^[9]).

First, we note that

${\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right) = \sum\limits_{q = 1}^{\infty}\frac{(1)_{q, \lambda}(\log_{\lambda}(1+z))^q}{(q-1)!q^k}$

$= \sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}(\log_{\lambda}(1+z))^{q+1}}{(q+1)^kq!}$

$= \sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}}{(q+1)^{k-1}}\frac{1}{(q+1)!}(\log_{\lambda}(1+z))^{q+1}$

$\begin{align} = \sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}}{(q+1)^{k-1}}\sum\limits_{r = q+1}^{\infty}S_{1, \lambda}(r, q+1)\frac{z^r}{r!}. \end{align}$

(2.3)

By making use of (2.1) and (2.3), we see that

$\frac{z}{\log_{\lambda}(1+z)}(1+z)^{x}{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right) \quad$

$= \frac{z}{\log_{\lambda}(1+z)}(1+z)^{x}\sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}}{(q+1)^{k-1}} \sum\limits_{r = q}^{\infty}\frac{S_{1, \lambda}(r+1, q+1)}{r+1}\frac{z^r}{r!}$

$= \sum\limits_{j = 0}^{\infty}b_{j, \lambda}(x)\frac{z^j}{j!}\sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}}{(q+1)^{k-1}}\sum\limits_{r = q}^{\infty} \frac{S_{1, \lambda}(r+1, q+1)}{r+1}\frac{z^r}{r!}$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\left(\sum\limits_{r = 0}^{j}\binom{j}{r}\sum\limits_{q = 0}^{r}\frac{(1)_{q+1, \lambda}}{(q+1)^{k-1}}\frac{S_{1, \lambda}(r+1, q+1)}{r+1}b_{j-r, \lambda}(x)\right)\frac{z^j}{j!}. \end{align}$

(2.4)

Therefore, by (2.3) and (2.4), we obtain the following theorem.

Theorem 2.1. For $k\in\mathbb Z$ and $j\geq 0$ , we have

$Pb_{j, \lambda}^{(k)}(x) = \sum\limits_{r = 0}^{j}\binom{j}{r}\sum\limits_{q = 0}^{r}\frac{(1)_{q+1, \lambda}}{(q+1)^{k-1}}\frac{S_{1, \lambda}(r+1, q+1)}{r+1} b_{j-r, \lambda}(x).$

Corollary 2.1. Putting $k = 1$ in Theorem 2.1 yields

$Pb_{j, \lambda}(x) = \sum\limits_{r = 0}^{j}\binom{j}{r}\sum\limits_{q = 0}^{r}\frac{(1)_{q+1, \lambda}S_{1, \lambda}(r+1, q+1)}{r+1}b_{j-r, \lambda}(x).$

Let $1\leq k\in\mathbb{Z}$ . For $s\in\mathbb C$ , the function $\chi_{k, \lambda}(s)$ is given as

$\begin{align} \chi_{k, \lambda}(s) = \frac{1}{\Gamma (s)}\int_{0}^{\infty}\frac{z^{s-1}}{\log_{\lambda}(1+z)}{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)dz. \end{align}$

(2.5)

From Eq (2.5), we have

$\chi_{k, \lambda}(s) = \frac{1}{\Gamma (s)}\int_{0}^{\infty}\frac{z^{s-1}}{\log_{\lambda}(1+z)}{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)dz$

$= \frac{1}{\Gamma (s)}\int_{0}^{1}\frac{z^{s-1}}{\log_{\lambda}(1+z)}{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)dz$

$\begin{align} +\frac{1}{\Gamma (s)}\int_{1}^{\infty}\frac{z^{s-1}}{\log_{\lambda}(1+z)}{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)dz. \end{align}$

(2.6)

For any $s\in\mathbb C$ , the second integral is absolutely convergent and thus, the second term on the r.h.s. vanishes at non-positive integers. That is,

$\begin{align} \lim\limits_{s\rightarrow -m}\left|\frac{1}{\Gamma (s)}\int_{1}^{\infty}\frac{z^{s-1}}{\log_{\lambda}(1+z)}{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)dz\right|\leq\frac{1}{\Gamma(-m)}M = 0. \end{align}$

(2.7)

On the other hand, the first integral in Eq (2.7), for $\Re(s) > 0$ can be written as

$\frac{1}{\Gamma (s)}\sum\limits_{r = 0}^{\infty}\frac{Pb_{r, \lambda}^{(k)}}{r!}\frac{1}{s+r},$

which defines an entire function of $s$ . Thus, we may include that $\chi_{k, \lambda}(s)$ can be continued to an entire function of $s$ .

Further, from (2.6) and (2.7), we obtain

$\chi_{k, \lambda}(-m) = \lim\limits_{s\rightarrow -m}\frac{1}{\Gamma (s)}\int_{0}^{1}\frac{z^{s-1}}{\log_{\lambda}(1+z)}{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)dz$

$= \lim\limits_{s\rightarrow -m}\frac{1}{\Gamma (s)}\int_{0}^{1}z^{s-1}\sum\limits_{r = 0}^{\infty}\frac{Pb_{r, \lambda}^{(k)}z^r}{r!}dz = \lim\limits_{s\rightarrow -m}\frac{1}{\Gamma (s)}\sum\limits_{r = 0}^{\infty}\frac{Pb_{r, \lambda}^{(k)}}{s+r}\frac{1}{r!}$

$\begin{align} = \cdots+0+\cdots+0+\lim\limits_{s\rightarrow -m}\frac{1}{\Gamma (s)}\frac{1}{s+m}\frac{Pb_{m, \lambda}^{(k)}}{m!}+0+0+\cdots \end{align}$

(2.8)

$= \lim\limits_{s\rightarrow -m}\frac{\left(\frac{\Gamma(1-s)\sin\pi s}{\pi}\right)}{s+m}\frac{Pb_{m, \lambda}^{(k)}}{m!} = \Gamma(1+m)\cos(\pi m)\frac{Pb_{m, \lambda}^{(k)}}{m!}$

$= (-1)^mPb_{m, \lambda}^{(k)}.$

In view of (2.8), we obtain the following theorem.

Theorem 2.2. Let $k\geq 1$ and $m\in\mathbb N\bigcup\{0\}$ , $s\in\mathbb C$ , we have

$\chi_{k, \lambda}(-m) = (-1)^mPb_{m, \lambda}^{(k)}.$

Using (1.8), we observe that

$\frac{\rm{d}}{\rm{dx}}{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+x)\right) = \frac{\rm{d}}{\rm{dx}}\sum\limits_{j = 1}^{\infty}\frac{(1)_{j, \lambda}(\log_{\lambda}(1+x))^{j}}{j^{k}(j-1)!}$

$\begin{align} = \frac{(1+x)^{\lambda-1}}{\log_{\lambda}(1+x)}\sum\limits_{j = 1}^{\infty}\frac{(1)_{j, \lambda}(\log_{\lambda}(1+x))^{j}}{j^{k-1}(j-1)!} = \frac{(1+x)^{\lambda-1}}{\log_{\lambda}(1+x)}{\rm Ei}_{k-1, \lambda}(\log_{\lambda}(1+x)). \end{align}$

(2.9)

Thus, by (2.9), for $k\geq 2$ , we get

${\rm Ei}_{k, \lambda}(\log_{\lambda}(1+x)) = \int_{0}^{x}\frac{(1+z)^{\lambda-1}}{\log(1+z)}{\rm Ei}_{k-1, \lambda}(\log_{\lambda}(1+z))dz$

$= \int_{0}^{x}{\underbrace {\frac{(1+z)^{\lambda-1}}{\log_{\lambda}(1+z)}\int_{0}^{z}\cdots\frac{(1+z)^{\lambda-1}}{\log_{\lambda}(1+z)} \int_{0}^{z}\frac{(1+z)^{\lambda-1}}{\log_{\lambda}(1+z)}}_{(k-2)- \mbox{times}}}dz\cdots dz$

$\times {\rm Ei}_{1, \lambda}(\log_{\lambda}(1+z))dz\cdots dz$

$\begin{align} = \int_{0}^{x}{\underbrace {\frac{(1+z)^{\lambda-1}}{\log_{\lambda}(1+z)}\int_{0}^{z}\cdots\frac{(1+z)^{\lambda-1}}{\log_{\lambda}(1+z)}\int_{0}^{z}\frac{(1+z)^{\lambda-1}}{\log_{\lambda}(1+z)}}_{(k-2)- \mbox{times}}}zdz\cdots dz. \end{align}$

(2.10)

From (2.1) and (2.10), we get

$\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}\frac{x^j}{j!} = \frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+x)\right)}{\log_{\lambda}(1+x)} = \frac{1}{\log_{\lambda}(1+x)}$

$\begin{align} \times\int_{0}^{x}{\underbrace {\frac{(1+z)^{\lambda-1}}{\log_{\lambda}(1+z)}\int_{0}^{z}\frac{(1+z)^{\lambda-1}}{\log_{\lambda}(1+z)}\cdots\int_{0}^{z}}_{(k-2)- \mbox{times}}}\frac{(1+z)^{\lambda-1}}{\log_{\lambda}(1+z)}zdz\cdots dz. \end{align}$

(2.11)

$= \frac{x}{\log_{\lambda}(1+x)}\sum\limits_{q = 0}^{\infty}\sum\limits_{q_1+\cdots+q_{k-1} = q}\binom{q}{q_1, \cdots, q_{k-1}}$

$\times \frac{b_{q_1, \lambda}(\lambda-1)}{q_1+1}\frac{b_{q_2, \lambda}(\lambda-1)}{q_1+q_2+1}\cdots\frac{b_{q_{k-1}, \lambda}(\lambda-1)}{q_1+\cdots+q_{k-1}+1}\frac{x^q}{q!}$

$= \sum\limits_{j = 0}^{\infty}\sum\limits_{q = 0}^{j}\binom{j}{q}\sum\limits_{q_1+\cdots+q_{k-1} = q}\binom{q}{q_1, \cdots, q_{k-1}}b_{j-q, \lambda}$

$\begin{align} \times \frac{b_{q_1, \lambda}(\lambda-1)}{q_1+1}\frac{b_{q_2, \lambda}(\lambda-1)}{q_1+q_2+1}\cdots\frac{b_{q_{k-1}, \lambda}(\lambda-1)}{q_1+\cdots+q_{k-1}+1}\frac{x^j}{j!}. \end{align}$

(2.12)

Therefore, by (2.12), we obtain the following theorem.

Theorem 2.3. For $j\in\mathbb N$ and $k\in\mathbb Z$ , we have

$Pb_{j, \lambda}^{(k)} = \sum\limits_{q = 0}^{j}\binom{j}{q}\sum\limits_{q_1+\cdots+q_{k-1} = q}\binom{q}{q_1, \cdots, q_{k-1}}b_{j-q, \lambda}$

$\times \frac{b_{q_1, \lambda}(\lambda-1)}{q_1+1}\frac{b_{q_2, \lambda}(\lambda-1)}{q_1+q_2+1}\cdots\frac{b_{q_{k-1}, \lambda}(\lambda-1)}{q_1+\cdots+q_{k-1}+1}.$

Corollary 2.2. Taking $k = 2$ in Theorem 2.3 yields

$Pb_{j, \lambda}^{(2)} = \sum\limits_{q = 0}^{j}\binom{j}{q}\frac{b_{q, \lambda}(\lambda-1)}{q+1}b_{j-q, \lambda}.$

Replacing $z$ by $e_{\lambda}(z)-1$ in (2.1), we get

$\sum\limits_{q = 0}^{\infty}Pb_{q, \lambda}^{(k)}(x)\frac{(e_{\lambda}(z)-1)^q}{q!} = \frac{{\rm Ei}_{k, \lambda}\left(z\right)}{z}e_{\lambda}^{x}(z)$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\frac{(x)_{j, \lambda}z^j}{j!}\sum\limits_{r = 0}^{\infty}\frac{(1)_{r+1, \lambda}z^r}{(r+1)^kr!} = \sum\limits_{j = 0}^{\infty} \left(\sum\limits_{r = 0}^{j}\binom{j}{r}\frac{(1)_{r+1, \lambda}(x)_{j-r, \lambda}}{(r+1)^k}\right)\frac{z^j}{j!}. \end{align}$

(2.13)

On the other hand,

$\sum\limits_{q = 0}^{\infty}Pb_{q, \lambda}^{(k)}(x)\frac{(e_{\lambda}(z)-1)^q}{q!} = \sum\limits_{q = 0}^{\infty}Pb_{q, \lambda}^{(k)}(x) \sum\limits_{j = q}^{\infty}S_{2, \lambda}(j, q)\frac{z^j}{j!}$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\left(\sum\limits_{q = 0}^{j}Pb_{q, \lambda}^{(k)}(x)S_{2, \lambda}(j, q)\right)\frac{z^j}{j!}. \end{align}$

(2.14)

In view of (2.13) and (2.14), we get the following theorem.

Theorem 2.4. For $k\in\mathbb Z$ and $j\geq 0$ , we have

$\sum\limits_{q = 0}^{j}Pb_{q, \lambda}^{(k)}(x)S_{2, \lambda}(j, q) = \sum\limits_{r = 0}^{j}\binom{j}{r}\frac{(1)_{r+1, \lambda}(x)_{j-r, \lambda}}{(r+1)^k}.$

By using (2.1), we get

$\sum\limits_{j = 1}^{\infty}\left[Pb_{j, \lambda}^{(k)}(x+1)-Pb_{j, \lambda}^{(k)}(x)\right]\frac{z^j}{j!} = \frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{\log_{\lambda}(1+z)}(1+z)^{x+1}-\frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{\log_{\lambda}(1+z)}(1+z)^{x}$

$= \frac{z{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{\log_{\lambda}(1+z)}(1+z)^{x} = \left(\frac{z}{\log_{\lambda}(1+z)}(1+z)^{x}\right) \left({\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)\right)$

$= \left(\sum\limits_{j = 0}^{\infty}b_{j, \lambda}(x)\frac{z^j}{j!}\right)\left(\sum\limits_{q = 1}^{\infty}\frac{(1)_{q, \lambda}(\log_{\lambda}(1+z))^q}{(q-1)!q^k}\right)$

$= \left(\sum\limits_{j = 0}^{\infty}b_{j, \lambda}(x)\frac{z^j}{j!}\right)\left(\sum\limits_{q = 1}^{\infty}\frac{(1)_{q, \lambda}(\log_{\lambda}(1+z))^{q}}{(q-1)!q^k}\right)$

$= \left(\sum\limits_{j = 1}^{\infty}b_{j, \lambda}(x)\frac{z^j}{j!}\right)\left(\sum\limits_{r = 1}^{\infty}\sum\limits_{q = 1}^{r}\frac{(1)_{q, \lambda}}{q^{k-1}}S_{1, \lambda}(r, q)\frac{z^r}{r!}\right)$

$\begin{align} = \sum\limits_{j = 1}^{\infty}\left(\sum\limits_{r = 1}^{j}\binom{j}{r}\sum\limits_{q = 1}^{r}\frac{(1)_{q, \lambda}}{q^{k-1}}S_{1, \lambda}(r, q)b_{j-r, \lambda}(x)\right)\frac{z^j}{j!}. \end{align}$

(2.15)

Therefore, by comparing the coefficients on both sides of (2.15), we obtain the following theorem.

Theorem 2.5. For $j\geq 0$ , we have

$Pb_{j, \lambda}^{(k)}(x+1)-Pb_{j, \lambda}^{(k)}(x) = \sum\limits_{r = 1}^{j}\binom{j}{r}\sum\limits_{q = 1}^{r}\frac{(1)_{q, \lambda}}{q^{k-1}}S_{1, \lambda}(r, q)b_{j-r, \lambda}(x).$

By making use of (1.3) and (2.1), we have

$\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}(x+\eta)\frac{z^j}{j!} = \frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{\log_{\lambda}(1+z)}(1+z)^{x+\eta}$

$= \frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{\log_{\lambda}(1+z)}(1+z)^{x}(1+z)^{\eta} = \left(\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}(x)\frac{z^j}{j!}\right) \left(\sum\limits_{q = 0}^{\infty}(\eta)_q\frac{z^q}{q!}\right)$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\left(\sum\limits_{q = 0}^{j}\binom{j}{q}Pb_{j-q, \lambda}^{(k)}(x)(\eta)_q\right)\frac{z^j}{j!}. \end{align}$

(2.16)

Therefore, by Eq (2.16), we obtain the following theorem.

Theorem 2.6. For $j\geq 0$ , we have

$Pb_{j, \lambda}^{(k)}(x+\eta) = \sum\limits_{q = 0}^{j}\binom{j}{q}Pb_{j-q, \lambda}^{(k)}(x)(\eta)_q.$

By using (2.1), we have

$\frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{\log_{\lambda}(1+z)} = \sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}\frac{z^j}{j!}$

${\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right) = \log_{\lambda}(1+z)\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}\frac{z^j}{j!}$

$\frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{z} = \frac{\log_{\lambda}(1+z)}{z}\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}\frac{z^j}{j!}$

$= \left(\sum\limits_{q = 0}^{\infty}D_{q, \lambda}\frac{t^q}{q!}\right)\left(\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}\frac{z^j}{j!}\right)$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\left(\sum\limits_{q = 0}^{j}\binom{j}{q}Pb_{j-q, \lambda}^{(k)}D_{q, \lambda}\right)\frac{z^j}{j!}. \end{align}$

(2.17)

On the other hand,

$\frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{z} = \frac{1}{z}\sum\limits_{q = 1}^{\infty}\frac{(1)_{q, \lambda}(\log_{\lambda}(1+z))^q}{(q-1)!q^k}$

$= \frac{1}{z}\sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}(\log_{\lambda}(1+z))^{q+1}}{q^k!(q+1)}$

$= \frac{1}{z}\sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}}{(q+1)^{k-1}}\frac{1}{(q+1)!}(\log_{\lambda}(1+z))^{q+1}$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\left(\sum\limits_{q = 0}^{j}\frac{(1)_{q+1, \lambda}}{(q+1)^{k-1}}\frac{S_{1, \lambda}(j+1, q+1)}{j+1}\right)\frac{z^j}{j!}. \end{align}$

(2.18)

Thus, by equations (2.17) and (2.18), we get the following theorem.

Theorem 2.7. For $j\geq 0$ , we have

$\sum\limits_{q = 0}^{j}\binom{j}{q}Pb_{j-q, \lambda}^{(k)}D_{q, \lambda} = \sum\limits_{q = 0}^{j}\frac{(1)_{q+1, \lambda}}{(q+1)^{k-1}}\frac{S_{1, \lambda}(j+1, q+1)}{j+1}.$

From (2.1), we have

$\sum\limits_{n = 0}^{\infty}Pb_{j, \lambda}^{(k)}(x)\frac{z^j}{j!} = \frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)} {\log_{\lambda}(1+z)}(1+z)^{x}$

$= \frac{{\rm Ei}_{k, \lambda}\left(\log_{\lambda}(1+z)\right)}{\log_{\lambda}(1+z)}e_{\lambda}^{x}(\log_{\lambda}(1+z))$

$= \sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}(x)\frac{z^j}{j!}\sum\limits_{q = 0}^{\infty}(x)_{q, \lambda}\sum\limits_{r = q}^{\infty}S_{1, \lambda}(r, q)\frac{z^r}{r!}$

$= \sum\limits_{j = 0}^{\infty}Pb_{j, \lambda}^{(k)}(x)\frac{z^j}{j!}\sum\limits_{r = 0}^{\infty}\sum\limits_{q = 0}^{r}(x)_{q, \lambda}S_{1, \lambda}(r, q)\frac{z^r}{r!}$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\left(\sum\limits_{r = 0}^{j}\binom{j}{r}Pb_{j-r, \lambda}^{(k)}(x)_{q, \lambda}S_{1, \lambda}(r, q)\right)\frac{z^j}{j!}. \end{align}$

(2.19)

Therefore, by comparing the coefficients on both sides of (2.19), we obtain the following theorem.

Theorem 2.8. For $j\geq 0$ , we have

$Pb_{j, \lambda}^{(k)}(x) = \sum\limits_{r = 0}^{j}\binom{j}{r}Pb_{j-r, \lambda}^{(k)}(x)_{q, \lambda}S_{1, \lambda}(r, q).$

3. The degenerate unipoly-Bernoulli polynomials of the second kind

Let $p$ be any arithmetic real or complex valued function defined on $\mathbb N$ . Kim-Kim ^[7] presented the unipoly function attached to polynomials $p(x)$ as

$\begin{align} u_k(x|p) = \sum\limits_{j = 1}^{\infty}\frac{p(j)}{j^k}x^n, (k\in\mathbb Z). \end{align}$

(3.1)

Moreover,

$\begin{align} u_k(x|1) = \sum\limits_{j = 1}^{\infty}\frac{x^j}{j^k} = {\rm Li}_{k}(x), \rm{(see \; [10, 14])}, \end{align}$

(3.2)

represent the known ordinary polylogarithm function.

Dolgy and Khan ^[3] introduced the degenerate unipoly function attached to polynomials $p(x)$ are considered as follows

$\begin{align} u_{k, \lambda}(x|p) = \sum\limits_{j = 1}^{\infty}p(j)\frac{(1)_{j, \lambda}x^j}{j^k}. \end{align}$

(3.3)

We see that

$\begin{align} u_{k, \lambda}\left(x|\frac{1}{\Gamma}\right) = {\rm Ei}_{k, \lambda}(x), \rm{{(see[14])}} \end{align}$

(3.4)

is the modified degenerate polyexponential function.

Now, we introduce the degenerate unipoly-Bernoulli polynomials of the second kind attached to polynomials $p(x)$ as

$\begin{align} \frac{u_{k, \lambda}\left(\log_{\lambda}(1+z)|p\right)}{\log_{\lambda}(1+z)}(1+z)^{x} = \sum\limits_{j = 0}^{\infty}Pb_{j, \lambda, p}^{(k)}(x)\frac{z^j}{j!}. \end{align}$

(3.5)

When $x = 0$ , $Pb_{j, \lambda, p}^{(k)} = Pb_{j, \lambda, p}^{(k)}(0)$ are called the degenerate unipoly-Bernoulli numbers of the second kind attached to $p$ .

If we take $p(j) = \frac{1}{\Gamma (j)}$ . Then, we have

$\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda, \frac{1}{\Gamma}}^{(k)}(x)\frac{z^j}{j!} = \frac{1}{\log_{\lambda}(1+z)}(1+z)^{x}u_{k, \lambda}\left(\log_{\lambda}(1+z)|\frac{1}{\Gamma}\right)$

$\begin{align} = \frac{1}{\log_{\lambda}(1+z)}(1+z)^{x}\sum\limits_{q = 1}^{\infty}\frac{(1)_{q, \lambda}(\log_{\lambda}(1+z))^q}{q^k(q-1)!}. \end{align}$

(3.6)

For $k = 1$ , we have

$\begin{align} \sum\limits_{j = 0}^{\infty}Pb_{j, \lambda, \frac{1}{\Gamma}}^{(1)}(x)\frac{z^j}{j!} = \frac{1}{\log_{\lambda}(1+z)}(1+z)^{x}\sum\limits_{q = 1}^{\infty} \frac{(1)_{q, \lambda}(\log_{\lambda}(1+z))^q}{q!} = \frac{z}{\log_{\lambda}(1+z)}(1+z)^{x}. \end{align}$

(3.7)

Thus, we have

$\begin{align} Pb_{j, \lambda, \frac{1}{\Gamma}}^{(1)}(x) = b_{j, \lambda}(x), (j\geq 0). \end{align}$

(3.8)

By making use of (1.12) and (3.3), we note that

$u_{k, \lambda}\left(\log_{\lambda}(1+z)|p\right) = \sum\limits_{q = 1}^{\infty}p(q)\frac{(1)_{q, \lambda}(\log_{\lambda}(1+z))^q}{q^k}$

$= \sum\limits_{q = 1}^{\infty}\frac{p(q)(1)_{q, \lambda}(\log_{\lambda}(1+z))^q}{q^k}\frac{q!}{q!}$

$= \sum\limits_{q = 1}^{\infty}\frac{p(q)(1)_{q, \lambda}q!}{q^k}\frac{(\log_{\lambda}(1+z))^q}{q!}$

$= \sum\limits_{q = 1}^{\infty}\frac{p(q)(1)_{q, \lambda}q!}{q^k}\sum\limits_{r = q}^{\infty}S_{1, \lambda}(r, q)\frac{z^r}{r!}$

$= \sum\limits_{r = 1}^{\infty}\left(\sum\limits_{q = 1}^{r}\frac{p(q)(1)_{q, \lambda}q!}{q^k}S_{1, \lambda}(r, q)\right)\frac{z^r}{r!}.$

Thus, we have the required result.

Lemma 3.1. For $k\in\mathbb Z$ , we have

$u_{k, \lambda}\left(\log_{\lambda}(1+z)|p\right) = \sum\limits_{r = 1}^{\infty}\left(\sum\limits_{q = 1}^{r}\frac{p(q)(1)_{q, \lambda}q!}{q^k}S_{1, \lambda}(r, q)\right)\frac{z^r}{r!}.$

Recalling from (3.5), we have

$\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda, p}^{(k)}(x)\frac{z^j}{j!} = \frac{1}{\log_{\lambda}(1+z)}(1+z)^{x} u_{k, \lambda}\left(\log_{\lambda}(1+z)|p\right)$

$= \frac{1}{\log_{\lambda}(1+z)}(1+z)^{x}\sum\limits_{q = 1}^{\infty}\frac{(1)_{q, \lambda}p(q)}{q^k}(\log_{\lambda}(1+z))^q$

$= \frac{1}{\log_{\lambda}(1+z)}(1+z)^{x}\sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}p(q+1)}{(q+1)^k}(\log_{\lambda}(1+z))^{q+1}$

$= \frac{1}{\log_{\lambda}(1+z)}(1+z)^{x}\sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}p(q+1)(q+1)!}{(q+1)^k}\sum\limits_{r = q+1}^{\infty}S_{r, \lambda}(r, q+1)\frac{z^r}{r!}$

$= \frac{z}{\log_{\lambda}(1+z)}(1+z)^{x}\sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}p(q+1)(q+1)!}{(q+1)^k}\sum\limits_{r = q}^{\infty}\frac{S_{1, \lambda}(r+1, q+1)}{r+1}\frac{z^r}{r!}$

$= \sum\limits_{j = 0}^{\infty}b_{j, \lambda}(x)\frac{z^j}{j!}\sum\limits_{r = 0}^{\infty} \left(\sum\limits_{q = 0}^{r}\frac{(1)_{q+1, \lambda}p(q+1)(q+1)!}{(q+1)^k}\frac{S_{1, \lambda}(r+1, q+1)}{r+1}\right)\frac{z^r}{r!}$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\left(\sum\limits_{r = 0}^{j}\sum\limits_{q = 0}^{r}\binom{j}{r}\frac{(1)_{q+1, \lambda}p(q+1)(q+1)!}{(q+1)^k} \frac{S_{1, \lambda}(r+1, q+1)}{r+1}b_{j-r, \lambda}(x)\right)\frac{z^j}{j!}. \end{align}$

(3.9)

Therefore, by comparing the coefficients on both sides of (3.9), we obtain the following theorem.

Theorem 3.1. For $j\geq 0$ and $k\in\mathbb Z$ . Then

$Pb_{j, \lambda, p}^{(k)}(x) = \sum\limits_{r = 0}^{j}\sum\limits_{q = 0}^{r}\binom{j}{r}\frac{(1)_{q+1, \lambda}p(q+1)(q+1)!}{(q+1)^k} \frac{S_{1, \lambda}(r+1, q+1)}{r+1}b_{j-r, \lambda}(x).$

Moreover,

$Pb_{j, \lambda, \frac{1}{\Gamma}}^{(k)}(x) = \sum\limits_{r = 0}^{j}\sum\limits_{q = 0}^{r}\binom{j}{r}\frac{b_{j-r, \lambda}(x)}{(q+1)^{k-1}} \frac{S_{1, \lambda}(r+1, q+1)}{r+1}.$

Using (3.5), we have

$\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda, p}^{(k)}(x)\frac{z^j}{j!} = \frac{1}{\log_{\lambda}(1+z)}u_{k, \lambda}(\log_{\lambda}(1+z)|p)(1+z)^{x}$

$= \frac{u_{k, \lambda}\left(\log_{\lambda}(1+z)|p\right)}{\log_{\lambda}(1+z)}\sum\limits_{j = 0}^{\infty}(x)_{j}\frac{z^j}{j!}$

$= \sum\limits_{i = 0}^{\infty}Pb_{i, \lambda, p}^{(k)}\frac{z^i}{i!}\sum\limits_{j = 0}^{\infty}(x)_{j}\frac{z^j}{j!}$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\left(\sum\limits_{i = 0}^{j}\binom{j}{i}Pb_{i, \lambda, p}^{(k)}(x)_{j-i}\right)\frac{z^j}{j!}. \end{align}$

(3.10)

Upon comparing the coefficients on both sides of Eq (3.10), we get the following theorem.

Theorem 3.2. For $j\geq 0$ and $k\in\mathbb Z$ . Then

$Pb_{j, \lambda, p}^{(k)}(x) = \sum\limits_{i = 0}^{j}\binom{j}{i}Pb_{i, \lambda, p}^{(k)}(x)_{j-i}.$

By making use of (1.11), (1.12) and (3.5), we have

$\sum\limits_{j = 0}^{\infty}Pb_{j, \lambda, p}^{(k)}\frac{z^j}{j!} = \frac{1}{\log_{\lambda}(1+z)}u_{k}\left(\log_{\lambda}(1+z)|p\right)$

$= \frac{1}{\log_{\lambda}(1+z)}\sum\limits_{q = 1}^{\infty}\frac{(1)_{q, \lambda}p(q)}{q^k}(\log_{\lambda}(1+z))^q$

$= \sum\limits_{q = 0}^{\infty}\frac{(1)_{q, \lambda}p(q+1)}{(q+1)^k}(\log_{\lambda}(1+z))^{q+1}$

$= \frac{z}{\log_{\lambda}(1+z)}\frac{\log_{\lambda}(1+z)}{z}\sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}p(q+1)q!}{(q+1)^k} \frac{(\log_{\lambda}(1+z))^{q}}{q!}$

$= \sum\limits_{j = 0}^{\infty}D_{j, \lambda}\frac{z^j}{j!}\sum\limits_{i = 0}^{\infty}b_{i, \lambda}\frac{z^i}{i!} \sum\limits_{q = 0}^{\infty}\frac{(1)_{q+1, \lambda}p(q+1)q!}{(q+1)^k}\sum\limits_{r = q}^{\infty}S_{1, \lambda}(r, q)\frac{z^r}{r!}$

$= \sum\limits_{j = 0}^{\infty}D_{j, \lambda}\frac{z^j}{j!}\sum\limits_{i = 0}^{\infty}b_{i, \lambda}\frac{z^i}{i!}\sum\limits_{r = 0}^{\infty} \sum\limits_{q = 0}^{r}\frac{(1)_{q+1, \lambda}p(q+1)q!}{(q+1)^k}S_{1, \lambda}(r, q)\frac{z^r}{r!}$

$= \sum\limits_{j = 0}^{\infty}\sum\limits_{i = 0}^{j}\binom{j}{i}D_{j-i, \lambda}b_{i, \lambda}\frac{z^j}{j!}\sum\limits_{r = 0}^{\infty} \sum\limits_{q = 0}^{r}\frac{(1)_{q+1, \lambda}p(q+1)q!}{(q+1)^k}S_{1, \lambda}(r, q)\frac{z^r}{r!}$

$\begin{align} = \sum\limits_{j = 0}^{\infty}\left(\sum\limits_{r = 0}^{j} \sum\limits_{q = 0}^{r}\sum\limits_{i = 0}^{j-r}\binom{j-r}{i}\binom{j}{r}D_{j-i-r, \lambda}b_{i, \lambda}\frac{(1)_{q+1, \lambda}p(q+1)q!}{(q+1)^k} S_{1, \lambda}(r, q)\right)\frac{z^j}{j!}. \end{align}$

(3.11)

Thus, by comparing the coefficients on both sides of (3.11), we obtain the following theorem.

Theorem 3.3. For $j\geq 0$ and $k\in\mathbb Z$ . Then

$Pb_{j, \lambda, p}^{(k)} = \sum\limits_{r = 0}^{j} \sum\limits_{q = 0}^{r}\sum\limits_{i = 0}^{j-r}\binom{j-r}{i}\binom{j}{r}D_{j-i-r, \lambda}b_{i, \lambda}\frac{(1)_{q+1, \lambda}p(q+1)q!}{(q+1)^k}S_{1, \lambda}(r, q).$

4. Numerical computations

In this section, certain numerical computations are done to calculate certain zeros of the degenerate poly-Bernoulli polynomials of the second kind and show some graphical representations. The first five members of $Pb_{j, \lambda}^{(k)}(x)$ are calculated and given as:

$\begin{align*} &Pb_{0, \lambda}^{(k)}(x) = 1, \\&Pb_{1, \lambda}^{(k)}(x) = \frac{1}{2}+x-\frac{1}{8 \log 3}-\frac{\log 81}{8 \log 3}, \\&Pb_{2, \lambda}^{(k)}(x) = \frac{1}{2}+x^2+\frac{10}{81 (\log 3)^2}+\frac{1}{8 \log 3}-\frac{x}{4 \log 3}-\frac{\log 81}{8 \log 3}-\frac{x \log 81}{4 \log 3}, \\&Pb_{3, \lambda}^{(k)}(x) = -\frac{1}{4}+2 x-\frac{3 x^2}{2}+x^3-\frac{5}{16 (\log 3)^3}-\frac{10}{27 (\log 3)^2}+\frac{10 x}{27 (\log 3)^2}\\& \quad -\frac{1}{4 \log 3}+\frac{3 x}{4 \log 3}-\frac{3 x^2}{8 \log 3}+\frac{\log 81}{16 \log 3}-\frac{3 x^2 \log 81}{8 \log 3}, \\&Pb_{4, \lambda}^{(k)}(x) = \frac{1}{2}-6 x+8 x^2-4 x^3+x^4+\frac{176}{125 (\log 3)^4}+\frac{15}{8 (\log 3)^3}-\frac{5 x}{4 (\log 3)^3}\\& \quad +\frac{110}{81 (\log 3)^2}-\frac{20 x}{9 (\log 3)^2}+\frac{20 x^2}{27 (\log 3)^2}+\frac{3}{4 \log 3}-\frac{11 x}{4 \log 3}\\& \quad +\frac{9 x^2}{4 \log 3}-\frac{x^3}{2 \log 3}-\frac{\log 81}{8 \log 3}+\frac{3 x^2 \log 81}{4 \log 3}-\frac{x^3 \log 81}{2 \log 3}. \end{align*}$

To show the behavior of $Pb_{j, \lambda}^{(k)}(x)$ , we display the graph $Pb_{j, \lambda}^{(k)}(x)$ for $k = 4$ and $\lambda = 3$ , this graph is presented in Figure 1.

Figure 1. Graph of

$Pb_{j, \lambda}^{(k)}(x)$ .

DownLoad: Full-Size Img PowerPoint

Next, the approximate solutions of $Pb_{j, \lambda}^{(k)}(x) = 0$ when $k = 4$ and $\lambda = 3$ , are calculated and listed in Table 1.

Table 1. Approximate solutions of

$Pb_{j, \lambda}^{(k)}(x) = 0$ .

$j$	Real zeros	Complex zeros
1	$0.11378$	-
2	$0.212959, \; 1.0146$	-
3	$0.468628, \; 0.788431, \; 2.08428$	-
4	$2.27482, \; 3.00114$	$0.589582 - 0.515659\; i, 0.589582 + 0.515659\; i$
5	$4.09322$	$0.470967 - 0.872952\; i, \; 0.470967 + 0.872952\; i,$
		$2.76687 - 0.464588\; i, \; 2.76687 + 0.464588\; i$
6	$4.47754, \; 4.94352$	$0.270509 - 1.2071\; i, \; 0.270509 + 1.2071\; i$
		$2.8603 - 1.06554\; i, \; 2.8603 + 1.06554\; i$
7	$6.12953$	$-0.00407237 - 1.52417\; i, \; -0.00407237 + 1.52417\; i,$
		$2.8544 - 1.67974\; i, \; 2.8544 +1.67974\; i$
		$4.98314 - 0.749479\; i, \; 4.98314 + 0.749479\; i$
8	-	$-0.344872 - 1.82511 \; i, \; -0.344872 + 1.82511 \; i,$
		$2.7537 - 2.30093\; i, \; 2.7537 + 2.30093\; i,$
		$5.21262 - 1.46596 \; i, \; 5.21262 + 1.46596 \; i,$
		$6.83367 - 0.248836 \; i, \; 6.83367 + 0.248836 \; i$

| Show Table

DownLoad: CSV

The zeros of $Pb_{j, \lambda}^{(k)}(x)$ for $\lambda \in \mathbb{C}, j = 12$ are plotted in Figure 2.

Figure 2. Zeros of

$Pb_{12, \lambda}^{(k)}(x)$ .

DownLoad: Full-Size Img PowerPoint

The stacking structure of approximate zeros of $Pb_{j, \lambda}^{(k)}(x) = 0$ for $\lambda = 4, j = 1, 2, ..., 12$ is given in Figure 3.

Figure 3. Stacking structure of zeros

$Pb_{j, \lambda}^{(k)}(x)$ .

DownLoad: Full-Size Img PowerPoint

5. Conclusions

In this article, we introduced the type 2 degenerate poly-Bernoulli polynomials of the second kind and derived many related interesting properties. Furthermore, we defined the degenerate unipoly Bernoulli polynomials of the second kind and established some considerable results. Finally, certain related beautiful zeros and graphs are shown.

Acknowledgments

The authors would like to express the gratitude to Deanship of Scientific Research at King Khalid University, Saudi Arabia for providing funding research group under the research grant number R G P.1/162/42.

Conflict of interest

The authors declare no conflict of interest.

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AIMS Mathematics

A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

Related Papers:

Abstract

1. Introduction

2. Type 2 degenerate poly-Bernoulli polynomials of the second kind

3. The degenerate unipoly-Bernoulli polynomials of the second kind

4. Numerical computations

5. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

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通讯作者: 陈斌, bchen63@163.com

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Abstract

1. Introduction

2. Type 2 degenerate poly-Bernoulli polynomials of the second kind

3. The degenerate unipoly-Bernoulli polynomials of the second kind

4. Numerical computations

5. Conclusions

Acknowledgments

Conflict of interest

References

AIMS Mathematics

A new family of degenerate poly-Bernoulli polynomials of the second kind with its certain related properties

Related Papers:

Abstract

1. Introduction

2. Type 2 degenerate poly-Bernoulli polynomials of the second kind

3. The degenerate unipoly-Bernoulli polynomials of the second kind

4. Numerical computations

5. Conclusions

Acknowledgments

Conflict of interest

References

This article has been cited by:

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通讯作者: 陈斌, bchen63@163.com

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Catalog

Abstract

1. Introduction

2. Type 2 degenerate poly-Bernoulli polynomials of the second kind

3. The degenerate unipoly-Bernoulli polynomials of the second kind

4. Numerical computations

5. Conclusions

Acknowledgments

Conflict of interest

References