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Analytical properties of type 2 degenerate poly-Bernoulli polynomials associated with their applications

Abstract

Recently, Kim et al. (Adv. Differ. Equ. 2020:168, 2020) considered the poly-Bernoulli numbers and polynomials resulting from the moderated version of degenerate polyexponential functions. In this paper, we investigate the degenerate type 2 poly-Bernoulli numbers and polynomials which are derived from the moderated version of degenerate polyexponential functions. Our degenerate type 2 degenerate poly-Bernoulli numbers and polynomials are different from those of Kim et al. (Adv. Differ. Equ. 2020:168, 2020) and Kim and Kim (Russ. J. Math. Phys. 26(1):40–49, 2019). Utilizing the properties of moderated degenerate poly-exponential function, we explore some properties of our type 2 degenerate poly-Bernoulli numbers and polynomials. From our investigation, we derive some explicit expressions for type 2 degenerate poly-Bernoulli numbers and polynomials. In addition, we also scrutinize type 2 degenerate unipoly-Bernoulli polynomials related to an arithmetic function and investigate some identities for those polynomials. In particular, we consider certain new explicit expressions and relations of type 2 degenerate unipoly-Bernoulli polynomials and numbers related to special numbers and polynomials. Further, some related beautiful zeros and graphical representations are displayed with the help of Mathematica.

Introduction

Carlitz [1, 2], Kim and Kim [14, 19, 2123], Kim et al. [26, 28, 29, 31], Jang et al. [7, 8], Muhiuddin et al. [3739], Khan et al. [1013], Sharma et al. [4143] introduced and studied various degenerate versions of special polynomials and numbers like degenerate Bernoulli polynomials, degenerate Euler polynomials, degenerate Daehee polynomials, degenerate Fubini polynomials, and degenerate Stirling numbers of the first and second kinds.

For \(\kappa \in \mathbb{Z}\), the modified degenerate polyexponential function [30] is specified by Kim and Kim to be

$$ {\mathrm{Ei}}_{\kappa ,\lambda }(\omega )=\sum_{\upsilon =1}^{ \infty } \frac{(1)_{\upsilon ,\lambda } \omega ^{\upsilon }}{(\upsilon -1)!\upsilon ^{\kappa }},\quad \bigl( | \omega | < 1\bigr). $$
(1.1)

Note that

$$ {\mathrm{Ei}}_{1,\lambda }(\omega )=\sum_{\upsilon =1}^{\infty } \frac{(1)_{\upsilon ,\lambda }\omega ^{\upsilon }}{\upsilon !} =e_{ \lambda }(\omega )-1. $$
(1.2)

The degenerate polyexponential functions were introduced by Kim and Kim as

$$ e_{\lambda }(\omega ,\delta |\kappa )=\sum_{\upsilon =0}^{ \infty } \frac{(1)_{\upsilon ,\lambda } \omega ^{\upsilon }}{\upsilon !(\upsilon +\delta )^{\kappa }} \quad \text{(see [21]),} $$
(1.3)

where \(\delta \in \mathbb{C}\) and \(\kappa \in \mathbb{N}\cup \{0\}\) with \(\Re (\delta )>0\).

From (1.3), we readily get

$$ e_{\lambda }(\omega ,\delta |0)=e_{\lambda }(\omega ),\qquad e_{\xi }( \omega ,1|1)= \frac{1}{\omega }\frac{1}{1+\xi }\bigl(e_{\lambda }(\omega )-1\bigr)+ \frac{\lambda }{1+\lambda }e_{\lambda }(\omega ). $$
(1.4)

We note here that

$$ \lim_{\lambda \rightarrow 0}e_{\lambda }(\omega ,1|1)= \frac{1}{\omega } \bigl(e^{\omega }-1\bigr)=e(\omega ,1|1), $$

which was defined by Hardy (see [5, 6]).

The degenerate poly-Genocchi polynomials [30] were considered by Kim and Kim and given as

$$ \frac{2{\mathrm{Ei}}_{\kappa ,\lambda } (\log _{\lambda }(1+z) )}{e_{\lambda }(z)+1}e_{ \lambda }^{\omega }(z) =\sum _{\upsilon =0}^{\infty }G_{ \upsilon ,\lambda }^{(\kappa )}(\omega ) \frac{z^{\upsilon }}{\upsilon !}\quad ( \kappa \in \mathbb{Z}). $$
(1.5)

When \(\omega =0\), \(G_{\upsilon ,\lambda }^{(\kappa )}=G_{\upsilon ,\lambda }^{(\kappa )}(0)\) are called degenerate poly-Genocchi numbers.

From (1.1) and (1.5), we see that \(G_{\upsilon ,\lambda }^{(1)}(\omega )=G_{\upsilon ,\lambda }(\omega )\), \(( \upsilon \geq 0)\) which are called degenerate Genocchi polynomials.

For \(\lambda \in \mathbb{R}\), the degenerate exponential function is defined as follows (see [1416, 1834]):

$$ e_{\lambda }^{\omega }(z)=(1+\lambda z)^{\frac{\omega }{\lambda }},\qquad e_{ \lambda }(z):=e_{\lambda }^{1}(z) =(1+\lambda z)^{\frac{1}{\lambda }} $$
(1.6)

and

$$ e_{\lambda }^{\omega }(z)=\sum_{\upsilon =0}^{\infty }( \omega )_{ \upsilon ,\lambda }\frac{z^{\upsilon }}{\upsilon !}, $$
(1.7)

where \((\omega )_{0,\lambda }=1\), \((\omega )_{\upsilon ,\lambda }=\omega ( \omega -\lambda )(\omega -2\lambda )\cdots (\omega -(\upsilon -1) \lambda )\), \((\upsilon \geq 1)\).

The degenerate Bernoulli polynomials considered by Carlitz [1, 2] are given by

$$ \frac{z}{e_{\lambda }(z)-1}e_{\lambda }^{\omega }(z) =\sum _{ \upsilon =0}^{\infty }\beta _{\upsilon ,\lambda }(\omega ) \frac{z^{\upsilon }}{\upsilon !}. $$
(1.8)

When \(\omega =0\), \(\beta _{\upsilon ,\lambda }=\beta _{\upsilon ,\lambda }(0)\) denotes degenerate Bernoulli numbers.

In 2019, Jang and Kim [7] introduced type 2 degenerate Bernoulli polynomials as follows:

$$ \frac{z}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}e_{ \lambda }^{\omega }(z)=\sum _{\upsilon =0}^{\infty }B_{\upsilon , \lambda } (\omega ) \frac{z^{\upsilon }}{\upsilon !}. $$
(1.9)

When \(\omega =0\), \(B_{\upsilon ,\lambda }=B_{\upsilon ,\lambda }(0)\) are type 2 degenerate Bernoulli numbers.

We note that

$$\begin{aligned} \lim_{\lambda \rightarrow 0} \frac{z}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}e_{ \lambda }^{\omega }(z) &=\sum_{\upsilon =0}^{\infty }\lim_{ \lambda \rightarrow 0}B_{\upsilon ,\lambda }( \omega ) \frac{z^{\upsilon }}{\upsilon !} \\ &=\frac{z}{e^{\frac{z}{2}}-e^{-\frac{z}{2}}}e^{\omega z}=\sum_{ \upsilon =0}^{\infty }B_{\upsilon }( \omega ) \frac{z^{\upsilon }}{\upsilon !}, \end{aligned}$$
(1.10)

are called type 2 Bernoulli polynomials \(B_{\upsilon }(\omega )\), \((\upsilon \geq 0)\).

The degenerate form of central Bell polynomials is given as (see [20])

$$ e^{\omega (e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z))}= \sum_{\upsilon =0}^{\infty } Bel_{\upsilon ,\lambda }^{(c)}( \omega )\frac{z^{\upsilon }}{\upsilon !}. $$
(1.11)

For \(\omega =1\), \(Bel_{\upsilon ,\lambda }^{(c)}=Bel_{\upsilon ,\lambda }^{(c)}(1)\) denotes degenerate central Bell numbers.

For \(\lambda \in \mathbb{R}\), Kim and Kim [15] defined the degenerate version of the logarithm function denoted by \(\log _{\lambda }(1+t)\) as follows:

$$ \log _{\lambda }(1+z)=\sum_{\upsilon =1}^{\infty } \lambda ^{ \upsilon -1}(1)_{\upsilon ,1/\lambda }\frac{z^{\upsilon }}{\upsilon !}, $$
(1.12)

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

$$ e_{\lambda }\bigl(\log _{\lambda }(z)\bigr)=\log _{\lambda } \bigl(e_{\lambda }(z)\bigr)=z. $$

It is noteworthy to mention that

$$ \lim_{\lambda \rightarrow 0}\log _{\lambda }(1+z)=\sum _{ \upsilon =1}^{\infty }(-1)^{\upsilon -1}\frac{z^{\upsilon }}{\upsilon !}= \log (1+z). $$

The degenerate Daehee polynomials of order r are defined by (see [28, 41])

$$ \biggl(\frac{\log _{\lambda }(1+z)}{z} \biggr)^{r}(1+z)^{\omega } = \sum _{\upsilon =0}^{\infty }D_{\upsilon ,\lambda }^{(r)}( \omega ) \frac{z^{\upsilon }}{\upsilon !}. $$
(1.13)

If \(\omega =0\), \(D_{\upsilon ,\lambda }^{(r)}=D_{\upsilon ,\lambda }^{(r)}(0) \), \(( \upsilon \geq 0)\) denotes degenerate Daehee numbers of order r.

For \(r=1\), (1.13) reduces to

$$ \frac{\log _{\lambda }(1+z)}{z}(1+z)^{\omega }=\sum_{\upsilon =0}^{ \infty }D_{\upsilon ,\lambda }( \omega )\frac{z^{\upsilon }}{\upsilon !}. $$
(1.14)

If \(\omega =0\), \(D_{\upsilon ,\lambda }=D_{\upsilon ,\lambda }(0)\), \((\upsilon \geq 0)\) denotes degenerate Daehee numbers.

The degenerate Bernoulli polynomials of the second kind are specified by

$$ \frac{z}{\log _{\lambda }(1+z)}(1+z)^{\omega }=\sum_{\upsilon =0}^{ \infty }b_{\upsilon ,\lambda }( \omega )\frac{z^{\upsilon }}{\upsilon !}\quad \text{(see [18])}. $$
(1.15)

When \(\omega =0\), \(b_{\upsilon ,\lambda }=b_{\upsilon ,\lambda }(0)\) (\(\upsilon \geq 0\)) denotes degenerate Bernoulli numbers of the second kind.

The degenerate form of first kind Stirling numbers is defined by

$$ \frac{1}{\kappa !}\bigl(\log _{\lambda }(1+z)\bigr)^{\kappa }=\sum _{ \upsilon =\kappa }^{\infty }S_{1,\lambda }(\upsilon ,\kappa ) \frac{z^{\upsilon }}{\upsilon !}\quad (\kappa \geq 0),\ {\text{(see [21, 30])}}. $$
(1.16)

Note here that \(\lim_{\lambda \rightarrow 0}S_{1,\lambda }(\upsilon ,\kappa )=S_{1}( \upsilon ,\kappa )\), where \(S_{1}(\upsilon ,\kappa )\) are first kind Stirling numbers given by

$$ \frac{1}{\kappa !}\bigl(\log (1+z)\bigr)^{\kappa }=\sum _{\upsilon = \kappa }^{\infty }S_{1}(\upsilon ,\kappa ) \frac{z^{\upsilon }}{\upsilon !}\quad ( \kappa \geq 0),\ {\text{(see [1--14, 17, 21, 26, 28])}}. $$

The degenerate form of second kind Stirling numbers is defined by

$$ \frac{1}{\kappa !} \bigl(e_{\lambda }(z)-1 \bigr)^{\kappa }=\sum _{\upsilon =\kappa }^{\infty }S_{2,\lambda }(\upsilon ,\kappa ) \frac{z^{\upsilon }}{\upsilon !} \quad \text{(see [17])}. $$
(1.17)

Here, \(\lim_{\lambda \rightarrow 0}S_{2,\lambda }(\upsilon ,\kappa )=S_{2}( \upsilon ,\kappa )\), where \(S_{2}(\upsilon ,\kappa )\) are second kind Stirling numbers given by

$$ \frac{1}{\kappa !}\bigl(e^{z}-1\bigr)^{\kappa }=\sum _{\upsilon =\kappa }^{ \infty }S_{2}(\upsilon ,\kappa ) \frac{z^{\upsilon }}{\upsilon !}\quad \text{(see [1--43])}. $$
(1.18)

The degenerate form of second kind central factorial polynomials [3] is given as

$$ \frac{1}{\kappa !} \bigl(e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{- \frac{1}{2}}(z) \bigr)^{\kappa }e_{\lambda }^{\omega }(z) =\sum _{j= \kappa }^{\infty }T_{2,\lambda }(j,\kappa |\omega ) \frac{z^{j}}{j!}\quad ( \kappa \geq 0). $$
(1.19)

If \(\omega =0\), \(T_{2,\lambda }(j,\kappa )=T_{2,\lambda }(j,\kappa |0)\) denotes second kind degenerate central factorial numbers.

This article aims to present type 2 degenerate poly-Bernoulli numbers and polynomials arising from moderated degenerate polyexponential functions. Certain explicit expressions for these numbers and polynomials are derived. Also, we introduce type 2 degenerate unipoly-Bernoulli numbers and polynomials by utilizing unipoly functions and show some basic properties of them.

Type 2 degenerate poly-Bernoulli polynomials and numbers

For \(\kappa \in \mathbb{Z}\), and utilizing the modified degenerate polyexponential functions, we consider type 2 degenerate poly-Bernoulli polynomials as

$$ \frac{{\mathrm{Ei}}_{\kappa ,\lambda } (\log _{\lambda }(1+z) )}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}e_{ \lambda }^{\omega }(z) =\sum _{\upsilon =0}^{\infty }\beta _{ \upsilon ,\lambda }^{(\kappa )}(\omega ) \frac{z^{\upsilon }}{\upsilon !}. $$
(2.1)

If \(\omega =0\), \(\beta _{\upsilon ,\lambda }^{(\kappa )}=\beta _{\upsilon ,\lambda }^{( \kappa )}(0)\) denotes type 2 degenerate poly-Bernoulli numbers.

For \(\kappa =1\), by using (2.1) and (1.2), we see that

$$ \frac{{\mathrm{Ei}}_{1,\lambda } (\log _{\lambda }(1+z) )}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}e_{ \lambda }^{\omega }(z) = \frac{z}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}e_{ \lambda }^{\omega }(z) =\sum_{\upsilon =0}^{\infty }B_{ \upsilon ,\lambda }(\omega ) \frac{z^{\upsilon }}{\upsilon !}\quad (\upsilon \geq 0), $$
(2.2)

where \(B_{\upsilon ,\lambda }(\omega )\) are called type 2 degenerate Bernoulli polynomials (see [3]).

First, we can write that

$$\begin{aligned} {\mathrm{Ei}}_{\kappa ,\lambda } \bigl(\log _{\lambda }(1+z) \bigr)&=\sum _{i=1}^{\infty } \frac{(1)_{i,\lambda }(\log _{\lambda }(1+z))^{i}}{(i-1)!i^{\kappa }} \\ &=\sum_{i=1}^{\infty }\frac{(1)_{i,\lambda }}{i^{\kappa -1}} \frac{(\log _{\lambda }(1+z))^{i}}{i!} \\ &=\sum_{i=1}^{\infty }\frac{(1)_{i,\lambda }}{i^{\kappa -1}} \sum _{\upsilon =i}^{\infty }S_{1,\lambda }(\upsilon ,i) \frac{z^{\upsilon }}{\upsilon !} \\ &=\sum_{\upsilon =1}^{\infty } \Biggl(\sum _{i=1}^{ \upsilon }\frac{(1)_{i,\lambda }}{i^{\kappa -1}} S_{1,\lambda }( \upsilon ,i) \Biggr)\frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(2.3)

By (2.3), we see that (2.1) is equal to

$$\begin{aligned} \frac{{\mathrm{Ei}}_{\kappa ,\lambda } (\log _{\lambda }(1+z) )}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} &=\frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{r=1}^{\infty } \Biggl(\sum_{i=1}^{r} \frac{(1)_{i,\lambda }}{i^{\kappa -1}}S_{1,\lambda }(r,i) \Biggr) \frac{z^{r}}{r!} \\ &=\frac{z}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{r=0}^{\infty } \Biggl( \sum_{i=1}^{r+1} \frac{(1)_{i,\lambda }}{i^{\kappa -1}} \frac{S_{1,\lambda }(r+1,i)}{r+1} \Biggr) \frac{z^{r}}{r!} \\ &=\sum_{\upsilon =0}^{\infty }B_{\upsilon ,\lambda } \frac{z^{\upsilon }}{\upsilon !}\sum_{r=0}^{\infty } \Biggl(\sum _{i=1}^{r+1} \frac{(1)_{i,\lambda }}{i^{\kappa -1}} \frac{S_{1,\lambda }(r+1,i)}{l+1} \Biggr)\frac{z^{r}}{r!} \\ &=\sum_{\upsilon =0}^{\infty } \Biggl(\sum _{r=0}^{ \upsilon }\binom{\upsilon }{r} \sum _{i=1}^{r+1} \frac{(1)_{i,\lambda }}{i^{\kappa -1}}\frac{S_{1,\lambda }(r+1,i)}{r+1}B_{ \upsilon -l,\lambda } \Biggr)\frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(2.4)

Therefore, by (2.1) and (2.4), we arrive at the following theorem.

Theorem 2.1

For \(\kappa \in \mathbb{Z}\) and \(\upsilon \geq 0\), we have

$$ \beta _{\upsilon ,\lambda }^{(\kappa )}=\sum_{r=0}^{\upsilon } \binom{\upsilon }{r}\sum_{i=1}^{r+1} \frac{(1)_{i,\lambda }}{i^{\kappa -1}} \frac{S_{1,\lambda }(r+1,i)}{l+1}B_{\upsilon -r,\lambda }. $$

Corollary 2.1

Putting \(k=1\) in Theorem  2.1 yields

$$ \beta _{\upsilon ,\lambda }=\sum_{r=0}^{\upsilon } \binom{\upsilon }{r}\sum_{i=1}^{r+1} \frac{S_{1,\lambda }(r+1,i)}{r+1}(1)_{i,\lambda } B_{\upsilon -r, \lambda }. $$

Remark 2.1

Letting λ to 0 in Theorem 2.1 leads to

$$ \beta _{\upsilon }^{(\kappa )}=\sum_{r=0}^{\upsilon } \binom{\upsilon }{r}\sum_{i=1}^{r+1} \frac{S_{1}(r+1,i)}{i^{\kappa -1}r+1} B_{\upsilon -r}\quad (\upsilon \geq 0). $$

Using (1.1), we have

$$\begin{aligned} \frac{\mathrm{{d}}}{\mathrm{{dz}}}{\mathrm{Ei}}_{\kappa ,\lambda }\bigl(\log _{\lambda }(1+z) \bigr)&= \frac{\mathrm{{d}}}{\mathrm{{dz}}} \sum_{\upsilon =1}^{\infty } \frac{(1)_{\upsilon ,\lambda } (\log _{\lambda }(1+z))^{\upsilon }}{(\upsilon -1)!\upsilon ^{\kappa }} \\ &=\frac{(1+z)^{\lambda -1}}{\log _{\lambda }(1+z)}\sum_{ \upsilon =1}^{\infty } \frac{(1)_{\upsilon ,\lambda } (\log _{\lambda }(1+z))^{\upsilon }}{(\upsilon -1)!\upsilon ^{\kappa -1}} \\ &=\frac{(1+z)^{\lambda -1}}{\log _{\lambda }(1+z)}{\mathrm{Ei}}_{\kappa -1, \lambda }\bigl(\log _{\lambda }(1+z) \bigr). \end{aligned}$$
(2.5)

By (2.5), for \(\kappa \geq 2\), we have

$$ {\mathrm{Ei}}_{\kappa ,\lambda }\bigl(\log _{\lambda }(1+z)\bigr)= \int _{0}^{z}{ \underbrace{\frac{(1+z)^{\lambda -1}}{\log _{\lambda }(1+z)} \int _{0}^{z}\cdots \frac{(1+z)^{\lambda -1}}{\log _{\lambda }(1+z)} \int _{0}^{z}}_{( \kappa -2)\text{-times}}} \frac{z(1+z)^{\lambda -1}}{\log _{\lambda }(1+z)} \,dz\cdots dz. $$
(2.6)

Then, from (2.1) and (2.6), we have

$$\begin{aligned} \sum_{\upsilon =0}^{\infty }\beta _{\upsilon ,\lambda }^{( \kappa )} \frac{z^{\upsilon }}{\upsilon !}&= \frac{1}{e_{\lambda }^{\frac{1}{2}}(z) -e_{\lambda }^{-\frac{1}{2}}(z)}{\mathrm{Ei}}_{\kappa ,\lambda } \bigl(\log _{ \lambda } (1+z) \bigr) \\ &=\frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \int _{0}^{z}{ \underbrace{\frac{(1+z)^{\lambda -1}}{\log _{\lambda } (1+z)} \int _{0}^{z}\cdots \frac{(1+z)^{\lambda -1}}{\log _{\lambda }(1+z)} \int _{0}^{z}}_{( \kappa -2)\text{-times}}} \frac{z(1+z)^{\lambda -1}}{\log _{\lambda }(1+z)} \,dz\cdots dz \\ &=\frac{z}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{i=0}^{\infty } \sum _{i_{1}+\cdots +i_{\kappa -1}=i} \binom{i}{i_{1}+\cdots +i_{\kappa -1}} \\ &\quad {}\times \frac{b_{i_{1},\lambda }(\lambda -1)}{i_{1}+1} \frac{b_{i_{2},\lambda }(\lambda -1)}{i_{1}+i_{2}+1}\cdots \frac{b_{i_{\kappa -1},\lambda }(\lambda -1)}{i_{1}+\cdots +i_{\kappa -1}+1} \frac{z^{i}}{i!} \\ &=\sum_{\upsilon =0}^{\infty }\sum _{i=0}^{\upsilon } \binom{\upsilon }{i}\sum _{i_{1}+\cdots +i_{\kappa -1}=i} \binom{i}{i_{1}+\cdots +i_{\kappa -1}} \\ &\quad {}\times \frac{b_{i_{1},\lambda }(\lambda -1)}{i_{1}+1} \frac{b_{i_{2},\lambda }(\lambda -1)}{i_{1}+i_{2}+1}\cdots \frac{b_{i_{\kappa -1},\lambda } (\lambda -1)}{i_{1}+\cdots +i_{\kappa -1}+1} B_{\upsilon -i,\lambda } \frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(2.7)

Therefore, by (2.7), we arrive at the following theorem.

Theorem 2.2

For \(\upsilon \geq 0\), then

$$\begin{aligned} \beta _{\upsilon ,\lambda }^{(\kappa )}&=\sum_{i=0}^{\upsilon } \binom{\upsilon }{i}\sum_{i_{1}+\cdots +i_{\kappa -1}=i} \binom{i}{i_{1}+\cdots +i_{\kappa -1}} \\ &\quad {}\times \frac{b_{i_{1},\lambda }(\lambda -1)}{i_{1}+1} \frac{b_{i_{2},\lambda }(\lambda -1)}{i_{1}+i_{2}+1}\cdots \frac{b_{i_{\kappa -1},\lambda }(\lambda -1)}{i_{1}+\cdots +i_{\kappa -1}+1} B_{\upsilon -i,\lambda }. \end{aligned}$$
(2.8)

By making use of (1.7) and (2.1), we note that

$$\begin{aligned} \sum_{\upsilon =0}^{\infty }\beta _{\upsilon ,\lambda }^{( \kappa )}( \omega )\frac{z^{\upsilon }}{\upsilon !} &= \frac{{\mathrm{Ei}}_{\kappa ,\lambda } (\log _{\lambda } (1+z) )}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}e_{ \lambda }^{\omega }(z) \\ &=\sum_{r=0}^{\infty }\beta _{r,\lambda }^{(\kappa )} \frac{z^{r}}{r!}\sum _{\upsilon =0}^{\infty }(\omega )_{ \upsilon ,\lambda } \frac{z^{\upsilon }}{\upsilon !} \\ & =\sum_{\upsilon =0}^{\infty } \Biggl(\sum _{r=0}^{ \upsilon }\binom{\upsilon }{r}\beta _{r,\lambda }^{(\kappa )} (\omega )_{ \upsilon -r,\lambda } \Biggr)\frac{z^{\upsilon }}{\upsilon !}, \end{aligned}$$
(2.9)

which on comparing the coefficients on both sides of the above equation yields the following theorem.

Theorem 2.3

For \(\upsilon \geq 0\), we have

$$ \beta _{\upsilon ,\lambda }^{(\kappa )}(\omega )=\sum_{r=0}^{ \upsilon } \binom{\upsilon }{r}\beta _{r,\lambda }^{(\kappa )} (\omega )_{ \upsilon -r,\lambda }. $$

Let \(\kappa \geq 1\) be an integer. For \(s\in \mathbb{C}\), the function \(\eta _{\kappa ,\lambda }(s)\) can be defined as

$$\begin{aligned} \eta _{\kappa ,\lambda }(s)&=\frac{1}{\Gamma (s)} \int _{0}^{\infty } \frac{z^{s-1}}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} { \mathrm{Ei}}_{\kappa ,\lambda } \bigl(\log _{\lambda }(1+z) \bigr)\,dz \\ &=\frac{1}{\Gamma (s)} \int _{0}^{1} \frac{z^{s-1}}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}{ \mathrm{Ei}}_{\kappa ,\lambda } \bigl(\log _{\lambda } (1+z) \bigr)\,dz \\ &\quad {}+\frac{1}{\Gamma (s)} \int _{1}^{\infty } \frac{z^{s-1}}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}{ \mathrm{Ei}}_{\kappa ,\lambda } \bigl(\log _{\lambda }(1+z) \bigr)\,dz. \end{aligned}$$
(2.10)

Here, for any \(s\in \mathbb{C}\), the second integral converges absolutely; hence, the second term on the r.h.s. vanishes at nonpositive integers, i.e.,

$$ \lim_{s\rightarrow -n} \biggl\vert \frac{1}{\Gamma (s)} \int _{1}^{\infty } \frac{z^{s-1}}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} { \mathrm{Ei}}_{\kappa ,\lambda } \bigl(\log _{\lambda }(1+z) \bigr)\,dz \biggr\vert \leq \frac{1}{\Gamma (-n)}N=0. $$
(2.11)

On the other hand, for \(\Re (s)>0\), we can write the first integral in (2.10) as

$$ \frac{1}{\Gamma (s)}\sum_{r=0}^{\infty } \frac{\beta _{r,\lambda }^{(\kappa )}}{r!}\frac{1}{s+r}, $$

which defines an entire function of s. Therefore, we may say that \(\eta _{\kappa ,\lambda }(s)\) can be continued to an entire function of s.

Further, from (2.10) and (2.11), we get

$$\begin{aligned} \eta _{\kappa ,\lambda }(-n)&=\lim_{s\rightarrow -n} \frac{1}{\Gamma (s)} \int _{0}^{1} \frac{z^{s-1}}{e_{\lambda }^{\frac{1}{2}} (z)-e_{\lambda }^{-\frac{1}{2}}(z)}{ \mathrm{Ei}}_{\kappa ,\lambda } \bigl( \log _{\lambda }(1+z) \bigr)\,dz \\ &=\lim_{s\rightarrow -n}\frac{1}{\Gamma (s)} \int _{0}^{1}z^{s-1}\sum _{r=0}^{\infty }\frac{B_{r,\lambda }^{(\kappa )}z^{r}}{r!}\,dz = \lim _{s\rightarrow -n}\frac{1}{\Gamma (s)}\sum_{r=0}^{ \infty } \frac{\beta _{r,\lambda }^{(\kappa )}}{s+r}\frac{1}{r!} \\ &=\cdots +0+\cdots +0+\lim_{s\rightarrow -n}\frac{1}{\Gamma (s)} \frac{1}{s+n}\frac{\beta _{n,\lambda }^{(\kappa )}}{n!}+0+0+\cdots \\ &=\lim_{s\rightarrow -n} \frac{ (\frac{\Gamma (1-s)\sin \pi s}{\pi } )}{s+n} \frac{\beta _{n,\lambda }^{(\kappa )}}{n!} =\Gamma (1+n)\cos (\pi n) \frac{\beta _{n,\lambda }^{(\kappa )}}{n!} \\ &=(-1)^{n}\beta _{n,\lambda }^{(\kappa )}. \end{aligned}$$
(2.12)

Therefore, by (2.12), we arrive at the following theorem.

Theorem 2.4

Let \(\kappa \geq 1\) and \(n\in \mathbb{N}\cup \{0\}\), \(s\in \mathbb{C}\). Then

$$ \eta _{\kappa ,\lambda }(-n)=(-1)^{n}\beta _{n,\lambda }^{(\kappa )}. $$

By making use of (1.2), we note that

$$\begin{aligned} {\mathrm{Ei}}_{1,\lambda } \bigl(\log _{\lambda }(1+z) \bigr)&=\sum _{i=1}^{ \infty }\frac{(1)_{i,\lambda }(\log _{\lambda }(1+z))^{i}}{i!} \\ &=\sum_{i=1}^{\infty } \frac{(1)_{i,\lambda }(\log _{\lambda }(1+z))^{i}}{i!} \\ &=\sum_{i=1}^{\infty }\sum _{\upsilon =i}^{\infty }(1)_{i, \lambda }S_{1,\lambda }(\upsilon ,i)\frac{z^{\upsilon }}{\upsilon !} \\ &=\sum_{\upsilon =1}^{\infty } \Biggl(\sum _{i=1}^{ \upsilon }(1)_{i,\lambda }S_{1,\lambda }(\upsilon ,i) \Biggr) \frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(2.13)

On the other hand,

$$ {\mathrm{Ei}}_{1,\lambda } \bigl(\log _{\lambda }(1+z) \bigr)=z. $$
(2.14)

Therefore, by (2.13) and (2.14), we arrive at the following theorem.

Theorem 2.5

For \(\upsilon \in \mathbb{N}\), then

$$ \sum_{i=1}^{\upsilon }(1)_{i,\lambda }S_{1,\lambda }( \upsilon ,i)= \delta _{\upsilon ,1}, $$

where \(\delta _{\upsilon ,\kappa }\) is the Kronecker delta.

From (2.1), we note that

$$ \frac{{\mathrm{Ei}}_{\kappa ,\lambda } (\log _{\lambda }(1+z) )}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} =\sum_{\upsilon =0}^{\infty }\beta _{\upsilon ,\lambda }^{( \kappa )}\frac{z^{\upsilon }}{\upsilon !}. $$
(2.15)

On replacing z with \(e_{\lambda }(z)-1\) in (2.15), we get

$$\begin{aligned} & \sum_{m=0}^{\infty }\beta _{m,\lambda }^{(\kappa )} \frac{(e_{\lambda }(z)-1)^{m}}{m!}\\ &\quad= \frac{{\mathrm{Ei}}_{\kappa ,\lambda }(z)}{e_{\lambda }^{\frac{1}{2}}(e_{\lambda }(z)-1)-e_{\lambda }^{-\frac{1}{2}}(e_{\lambda }(z)-1)} \\ &\quad= \frac{e_{\lambda }(z)-1}{e_{\lambda }^{\frac{1}{2}}(e_{\lambda }(z)-1)-e_{\lambda }^{-\frac{1}{2}}(e_{\lambda }(z)-1)} \frac{z}{e_{\lambda }(z)-1}\frac{{\mathrm{Ei}}_{\kappa ,\lambda }(z)}{z} \\ &\quad=\sum_{i=0}^{\infty }B_{i,\lambda } \frac{1}{i!}\bigl(e_{\lambda }(z)-1\bigr)^{i} \sum _{j=0}^{\infty }B_{j,\lambda }\frac{z^{j}}{j!} \sum _{\upsilon =0}^{\infty } \frac{(1)_{\upsilon +1,\lambda }z^{\upsilon }}{\upsilon !(\upsilon +1)^{\kappa }} \\ &\quad=\sum_{i=0}^{\infty }B_{i,\lambda }\sum _{l=i}^{\infty }S_{2, \lambda }(l,i) \frac{z^{l}}{l!}\sum_{j=0}^{\infty } B_{j, \lambda } \frac{z^{j}}{j!} \sum_{\upsilon =0}^{\infty } \frac{(1)_{\upsilon +1,\lambda }z^{\upsilon }}{\upsilon !(\upsilon +1)^{\kappa }} \\ &\quad=\sum_{l=0}^{\infty }\sum _{i=0}^{l}B_{i,\lambda }S_{2, \lambda }(l,i) \frac{z^{l}}{l!} \sum_{j=0}^{\infty }B_{j, \lambda } \frac{z^{j}}{j!} \sum_{\upsilon =0}^{\infty } \frac{(1)_{\upsilon +1,\lambda }z^{\upsilon }}{\upsilon !(\upsilon +1)^{\kappa }} \\ &\quad=\sum_{j=0}^{\infty } \Biggl(\sum _{l=0}^{j}\sum_{i=0}^{l} \binom{j}{l}B_{i,\lambda }S_{2,\lambda } (l,i)B_{j-l, \lambda } \Biggr) \frac{z^{j}}{j!} \times \sum_{\upsilon =0}^{ \infty } \frac{(1)_{\upsilon +1,\lambda }z^{\upsilon }}{\upsilon !(\upsilon +1)^{\kappa }} \\ &\quad=\sum_{\upsilon =0}^{\infty } \Biggl(\sum _{j=0}^{ \upsilon }\sum_{l=0}^{j} \sum_{i=0}^{l}\binom{j}{l} \binom{\upsilon }{j} B_{i,\lambda } S_{2,\lambda }(l,i)B_{j-l,\lambda } \frac{(1)_{\upsilon -j+1,\lambda }}{(\upsilon -j+1)^{\kappa }} \Biggr) \frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(2.16)

On the other hand,

$$\begin{aligned} \sum_{m=0}^{\infty }\beta _{m,\lambda }^{(\kappa )} \frac{(e_{\lambda }(z)-1)^{m}}{m!}&=\sum_{m=0}^{\infty }\beta _{m, \lambda }^{(\kappa )} \sum_{\upsilon =m}^{\infty }S_{2,\lambda }( \upsilon ,m)\frac{z^{\upsilon }}{\upsilon !} \\ &=\sum_{\upsilon =0}^{\infty } \Biggl(\sum _{m=0}^{ \upsilon }\beta _{m,\lambda }^{(\kappa )}S_{2,\lambda }( \upsilon ,m) \Biggr) \frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(2.17)

Therefore, by (2.16) and (2.17), we arrive at the following theorem.

Theorem 2.6

For \(\upsilon \in \mathbb{N}\), we have

$$ \sum_{m=0}^{\upsilon }\beta _{m,\lambda }^{(\kappa )}S_{2, \lambda }( \upsilon ,m)=\sum_{j=0}^{\upsilon }\sum _{l=0}^{j} \sum_{i=0}^{l} \binom{j}{l}\binom{\upsilon }{j}B_{i,\lambda } S_{2, \lambda }(l,i)B_{j-l,\lambda } \frac{(1)_{\upsilon -j+1,\lambda }}{(\upsilon -j+1)^{\kappa }}. $$

The higher-order type 2 degenerate Bernoulli polynomials of order \(r\in \mathbb{N}\) are defined by (see [7])

$$ \biggl( \frac{z}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \biggr)^{r}e_{\lambda }^{\omega }(z) =\sum_{\upsilon =0}^{ \infty }B_{\upsilon ,\lambda }^{(r)}( \omega ) \frac{z^{\upsilon }}{\upsilon !}. $$
(2.18)

For \(\omega =0\), \(B_{\upsilon ,\lambda }^{(r)}(0)=B_{\upsilon ,\lambda }^{(r)}\) denotes higher-order type 2 degenerate Bernoulli numbers of order \(r\in \mathbb{N}\).

From (1.18) and (2.18), we get

$$ \binom{\upsilon +\kappa }{\kappa }B_{\upsilon ,\lambda }^{(-\kappa )}=T_{2, \lambda }(\upsilon + \kappa ,\kappa ), $$

where υ, κ are nonnegative positive integers.

Replacing z with \(\log (1+z)\) in (2.18), we get

$$\begin{aligned} \sum_{m=0}^{\infty }B_{m,\lambda }^{(r)} \frac{(\log (1+z))^{m}}{m!} &= \biggl( \frac{\log (1+z)}{(1+\lambda\log(1+z))^{\frac{1}{2\lambda}}-(1+\lambda\log(1+z))^{-\frac{1}{2\lambda}}} \biggr)^{r} \\ &=\sum_{\upsilon =0}^{\infty } D^{*(r)}_{\upsilon,\lambda}\frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(2.19)

On the other hand,

$$\begin{aligned} \sum_{m=0}^{\infty }B_{m,\lambda }^{(r)} \frac{(\log (1+z))^{m}}{m!}&=\sum_{m=0}^{\infty }B_{m, \lambda }^{(r)} \sum_{\upsilon =m}^{\infty }S_{1 }( \upsilon ,m)\frac{z^{\upsilon }}{\upsilon !} \\ &=\sum_{\upsilon =0}^{\infty } \Biggl(\sum _{m=0}^{ \upsilon }B_{m,\lambda }^{(r)}S_{1 }( \upsilon ,m) \Biggr) \frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(2.20)

Therefore, by (2.19) and (2.20), we arrive at the following theorem.

Theorem 2.7

For \(\upsilon \geq 0\), we have

$$ D^{*(r)}_{\upsilon,\lambda} =\sum_{m=0}^{\upsilon }B^{(r)}_{m,\lambda}S_{1}(\upsilon,m). $$

Now, we introduce type 2 degenerate central poly-Bell polynomials as

$$ {\mathrm{Ei}}_{\kappa ,\lambda } (\omega \bigl(e_{\lambda }^{\frac{1}{2}}(z)-e_{ \lambda }^{-\frac{1}{2}}(z) \bigr) =\sum_{\upsilon =1}^{ \infty }Bel_{\upsilon ,\lambda }^{(\kappa ,c)}( \omega ) \frac{z^{\upsilon }}{\upsilon !},\qquad Bel_{\upsilon ,\lambda }^{(\kappa ,c)}( \omega )=0\quad (\kappa \in \mathbb{Z}). $$
(2.21)

When \(\omega =1\), \(Bel_{\upsilon ,\lambda }^{(\kappa ,c)}=Bel_{\upsilon ,\lambda }^{( \kappa ,c)}(1)\) are called type 2 degenerate poly-Bell numbers.

From (2.21), we note that

$$\begin{aligned} {\mathrm{Ei}}_{\kappa ,\lambda } (\omega \bigl(e_{\lambda }^{\frac{1}{2}}(z)-e_{ \lambda }^{-\frac{1}{2}}(z) \bigr) &=\sum_{m=1}^{\infty } \frac{\omega ^{m}(e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z))^{m}}{(m-1)!n^{\kappa }} \\ &=\sum_{m=1}^{\infty } \frac{\omega ^{m}(e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z))^{m}}{(m-1)!m^{\kappa }} \\ &=\sum_{m=1}^{\infty }\frac{\omega ^{m}}{m^{\kappa -1}}\sum _{\upsilon =m}^{\infty }T_{2,\lambda }(\upsilon ,m) \frac{z^{\upsilon }}{\upsilon !} \\ & =\sum_{\upsilon =1}^{\infty } \Biggl( \sum_{m=1}^{\upsilon } \frac{\omega ^{m}}{m^{\kappa -1}}T_{2,\lambda }( \upsilon ,m) \Biggr) \frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(2.22)

Thus, by (2.21) and (2.22), we obtain the following theorem.

Theorem 2.8

For \(\kappa \in \mathbb{Z}\) and \(\upsilon \in \mathbb{N}\), we have

$$ Bel_{\upsilon ,\lambda }^{(\kappa ,c)}(\omega )=\sum_{m=1}^{ \upsilon } \frac{\omega ^{m}}{m^{\kappa -1}}T_{2,\lambda }(\upsilon ,m). $$

Type 2 degenerate unipoly-Bernoulli polynomials and numbers

The unipoly function attached to polynomials \(p(\omega )\) was defined by Kim and Kim [14] as

$$ u_{\kappa }(\omega |p)=\sum_{\upsilon =1}^{\infty } \frac{p(\upsilon )}{\upsilon ^{\kappa }}\omega ^{\upsilon }\quad (\kappa \in \mathbb{Z}), $$
(3.1)

where p denotes any arithmetic real or complex-valued function defined on \(\mathbb{N}\).

Moreover,

$$ u_{\kappa }(\omega |1)=\sum_{\upsilon =1}^{\infty } \frac{\omega ^{\upsilon }}{\upsilon ^{\kappa }}={\mathrm{Li}}_{\kappa }(\omega ),\quad \text{(see [9])} $$
(3.2)

is the ordinary polylogarithm function.

Dolgy and Khan [4] introduced the degenerate unipoly function attached to polynomials \(p(\omega )\) as follows:

$$ u_{\kappa ,\lambda }(\omega |p)=\sum_{i=1}^{\infty }p(i) \frac{(1)_{i,\lambda } \omega ^{i}}{i^{\kappa }}. $$
(3.3)

It is worthy to see that

$$ u_{\kappa ,\lambda } \biggl(\omega \Big|\frac{1}{\Gamma } \biggr)={ \mathrm{Ei}}_{ \kappa ,\lambda }(\omega ),\quad \text{{(see [30])}} $$
(3.4)

is the moderated degenerate polyexponential function.

Now, we define type 2 degenerate unipoly-Bernoulli polynomials which are given by the generating function as follows:

$$ \frac{u_{\kappa ,\lambda } (\log _{\lambda }(1+z)|p )}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}e_{ \lambda }^{\omega }(z) =\sum _{r=0}^{\infty }\beta _{r,\lambda ,p}^{( \kappa )}(\omega )\frac{z^{r}}{r!}. $$
(3.5)

For \(\omega =0\), \(\beta _{r,\lambda ,p}^{(\kappa )}=\beta _{r,\lambda ,p}^{(\kappa )}(0)\) denotes type 2 degenerate unipoly-Bernoulli numbers attached to p.

By (3.5), we see that

$$\begin{aligned} \sum_{\upsilon =0}^{\infty }\beta _{\upsilon ,\lambda , \frac{1}{\Gamma }}^{(\kappa )} \frac{z^{\upsilon }}{\upsilon !} &= \frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}u_{ \kappa ,\lambda } \biggl(\log _{\lambda }(1+z)\Big|\frac{1}{\Gamma } \biggr) \\ &=\frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{r=1}^{\infty } \frac{(1)_{r,\lambda }(\log _{\lambda } (1+z))^{r}}{r^{\kappa }(r-1)!} \\ &=\frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}u_{ \kappa ,\lambda } \bigl(\log _{\lambda }(1+z) \bigr) = \sum_{ \upsilon =0}^{\infty }\beta _{\upsilon ,\lambda }^{(\kappa )} \frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(3.6)

Thus, by (3.6), we have

$$ \beta _{\upsilon ,\lambda ,\frac{1}{\Gamma }}^{(\kappa )}=\beta _{ \upsilon ,\lambda }^{(\kappa )}. $$
(3.7)

By making use of (3.5), we see that

$$\begin{aligned} \sum_{\upsilon =0}^{\infty }\beta _{\upsilon ,\lambda ,p}^{( \kappa )} \frac{z^{\upsilon }}{\upsilon !}& = \frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}u_{ \kappa ,\lambda } \bigl(\log _{\lambda }(1+z)\bigr)|p ) \\ &=\frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{r=1}^{\infty } \frac{p(r)(1)_{r,\lambda }}{r^{\kappa }}\bigl( \log _{\lambda }(1+z)\bigr))^{r} \\ &=\frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{r=1}^{\infty } \frac{p(r)(1)_{r,\lambda }}{r^{\kappa }}\bigl( \log _{\lambda } (1+z)\bigr)^{r} \frac{r!}{r!} \\ &=\frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{r=1}^{\infty } \frac{p(r)(1)_{r,\lambda }r!}{r^{\kappa }} \sum_{i=r}^{\infty }S_{1,\lambda }(i,r) \frac{z^{i}}{i!} \\ &=\frac{z}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{i=0}^{\infty } \sum _{r=1}^{i+1} \frac{p(r)(1)_{r,\lambda }r!}{r^{\kappa }(i+1)}S_{1,\lambda }(i+1,r) \frac{z^{i}}{i!} \\ &= \Biggl(\sum_{j=0}^{\infty }B_{j,\lambda } \frac{z^{j}}{j!} \Biggr) \Biggl(\sum_{i=0}^{\infty } \sum_{r=1}^{i+1} \frac{p(r)(1)_{r,\lambda }r!}{r^{\kappa }(i+1)}S_{1,\lambda }(i+1,r) \frac{z^{i}}{i!} \Biggr) \\ &=\sum_{j=0}^{\infty } \Biggl(\sum _{i=0}^{j}\sum_{r=1}^{i+1} \binom{j}{i} \frac{p(r)(1)_{r,\lambda }r!}{r^{\kappa }(i+1)}S_{1,\lambda }(i+1,r)B_{j-i, \lambda } \Biggr)\frac{z^{j}}{j!}, \end{aligned}$$
(3.8)

which yields the following theorem.

Theorem 3.1

Let j be a nonnegative integer and \(\kappa \in \mathbb{Z}\). Then

$$ \beta _{j,\lambda ,p}^{(\kappa )}=\sum_{i=0}^{j} \sum_{r=1}^{i+1}\binom{j}{i} \frac{p(r)(1)_{r,\lambda }r!}{r^{\kappa }(i+1)}S_{1,\lambda }(i+1,r)B_{j-i, \lambda }. $$

Recalling from (3.5), we have

$$\begin{aligned} \sum_{j=0}^{\infty }\beta _{j,\lambda ,p}^{(\kappa )}( \omega ) \frac{z^{j}}{j!} &= \frac{u_{\kappa ,\lambda } (\log _{\lambda }(1+z))|p )}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}e_{ \lambda }^{\omega }(z) \\ &= \frac{u_{\kappa ,\lambda } (\log _{\lambda }(1+z))|p )}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{s=0}^{\infty }( \omega )_{s,\lambda }\frac{z^{s}}{s!} \\ &=\sum_{j=0}^{\infty }\beta _{j,\lambda ,p}^{(\kappa )} \frac{z^{j}}{j!}\sum_{s=0}^{\infty }(\omega )_{s,\lambda } \frac{z^{s}}{s!} \\ &=\sum_{j=0}^{\infty } \Biggl(\sum _{s=0}^{j} \binom{j}{s}\beta _{j-s,\lambda ,p}^{(\kappa )}(\omega )_{s,\lambda } \Biggr) \frac{z^{j}}{j!}. \end{aligned}$$
(3.9)

By Eq. (3.9), we get the following theorem.

Theorem 3.2

Let j be a nonnegative integer and \(\kappa \in \mathbb{Z}\). Then

$$ \beta _{j,\lambda ,p}^{(\kappa )}(\omega )=\sum_{s=0}^{j} \binom{j}{s}\beta _{j-s,\lambda ,p}^{(\kappa )}(\omega )_{s,\lambda }. $$

From (3.5), we have

$$\begin{aligned} \sum_{j=0}^{\infty }\beta _{j,\lambda ,p}^{(\kappa )}( \omega ) \frac{z^{j}}{j!} &= \frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}u_{ \kappa ,\lambda }\bigl(\log _{\lambda }(1+z)|p\bigr) \bigl(e_{\lambda }(z)-1+1 \bigr)^{\omega } \\ &= \frac{u_{\kappa ,\lambda }(\log _{\lambda }(1+z)|p)}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{i=0}^{\infty } ( \omega )_{i} \frac{(e_{\lambda }(z)-1)^{i}}{i!} \\ &=\sum_{j=0}^{\infty }\beta _{j,\lambda ,p}^{(\kappa )} \frac{z^{j}}{j!}\sum_{i=0}^{\infty }(\omega )_{i}\sum_{q=i}^{\infty }S_{2,\lambda }(q,i) \frac{z^{q}}{q!} \\ &=\sum_{j=0}^{\infty }\beta _{j,\lambda ,p}^{(\kappa )} \frac{z^{j}}{j!}\sum_{q=0}^{\infty }\sum _{i=0}^{q}( \omega )_{i}S_{2,\lambda }(q,i) \frac{z^{q}}{q!} \\ &=\sum_{j=0}^{\infty } \Biggl(\sum _{q=0}^{j}\sum_{i=0}^{q} \binom{j}{q}(\omega )_{i}S_{2,\lambda }(q,i)\beta _{j-q, \lambda ,p}^{(\kappa )} \Biggr)\frac{z^{j}}{j!}, \end{aligned}$$
(3.10)

which yields the following theorem.

Theorem 3.3

Let j be a nonnegative integer and \(\kappa \in \mathbb{Z}\). Then

$$ \beta _{j,\lambda ,p}^{(\kappa )}(\omega )=\sum_{q=0}^{j} \sum_{i=0}^{q}\binom{j}{q}(\omega )_{i}S_{2,\lambda }(q,i) \beta _{j-q,\lambda ,p}^{(\kappa )}. $$

Using (3.5), we have

$$\begin{aligned} \sum_{\upsilon =0}^{\infty }\beta _{\upsilon ,\lambda ,p}^{( \kappa )} \frac{z^{\upsilon }}{\upsilon !} &= \frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)}u_{ \kappa ,\lambda } \bigl(\log _{\lambda }(1+z)|p \bigr) \\ &=\frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{m=1}^{\infty } \frac{p(m)(1)_{m,\lambda }}{m^{\kappa }}\bigl( \log _{\lambda }(1+z)\bigr)^{m} \\ &=\frac{1}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{m=0}^{\infty } \frac{p(m+1)(1)_{m+1,\lambda }}{(m+1)^{\kappa }}\bigl(\log _{\lambda }(1+z)\bigr)^{m+1} \\ &= \frac{\log _{\lambda }(1+z)}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{m=0}^{\infty } \frac{p(m+1)(1)_{m+1,\lambda }m!}{(m+1)^{\kappa }} \frac{(\log _{\lambda }(1+z))^{m}}{m!} \\ &= \frac{\log _{\lambda }(1+z)}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{m=0}^{\infty } \frac{p(m+1)(1)_{m+1,\lambda }m!}{(m+1)^{\kappa }}\sum_{l=m}^{ \infty }S_{1,\lambda }(l,m) \frac{z^{l}}{l!} \\ &=\frac{\log _{\lambda }(1+z)}{z} \frac{z}{e_{\lambda }^{\frac{1}{2}}(z)-e_{\lambda }^{-\frac{1}{2}}(z)} \sum_{l=0}^{\infty } \sum_{m=0}^{l} \frac{p(m+1)(1)_{m+1,\lambda }m!}{(m+1)^{\kappa }}S_{1,\lambda }(l,m) \frac{z^{l}}{l!} \\ &=\sum_{\upsilon =0}^{\infty }D_{\upsilon ,\lambda } \frac{z^{\upsilon }}{\upsilon !}\sum_{j=0}^{\infty }B_{j,\lambda } \frac{z^{j}}{j!} \sum_{l=0}^{\infty } \sum _{m=0}^{l} \frac{p(m+1)(1)_{m+1,\lambda }m!}{(m+1)^{\kappa }}S_{1,\lambda }(l,m) \frac{z^{l}}{l!} \\ &=\sum_{\upsilon =0}^{\infty }\sum _{j=0}^{\upsilon } \binom{\upsilon }{j}D_{\upsilon -j,\lambda }B_{j,\lambda } \frac{z^{\upsilon }}{\upsilon !} \sum_{l=0}^{\infty } \sum _{m=0}^{l}\frac{p(m+1)(1)_{m+1,\lambda }m!}{(m+1)^{\kappa }}S_{1, \lambda }(l,m) \frac{z^{l}}{l!} \\ &=\sum_{\upsilon =0}^{\infty } \Biggl(\sum _{l=0}^{ \upsilon } \sum_{m=0}^{l} \sum_{j=0}^{\upsilon -l} \binom{\upsilon -l}{j} \binom{\upsilon }{l} \\ &\quad {}\times D_{\upsilon -j-l,\lambda }B_{j, \lambda }\frac{p(m+1)(1)_{m+1,\xi }m!}{(m+1)^{\kappa }} S_{1,\lambda }(l,m) \Biggr)\frac{z^{\upsilon }}{\upsilon !}. \end{aligned}$$
(3.11)

Therefore, by (3.5) and (3.11), we obtain the following theorem.

Theorem 3.4

For \(\upsilon \geq 0\) and \(\kappa \in \mathbb{Z}\), we have

$$ \beta _{\upsilon ,\lambda ,p}^{(\kappa )}=\sum_{l=0}^{ \upsilon } \sum_{m=0}^{l}\sum _{j=0}^{\upsilon -l} \binom{\upsilon -l}{j}\binom{\upsilon }{l}D_{\upsilon -j-l,\lambda }B_{j, \lambda } \frac{p(m+1)(1)_{m+1,\lambda }m!}{(m+1)^{\kappa }}S_{1,\lambda }(l,m). $$

Numerical computations

In this section, some numerical computations are done to calculate certain zeros of type 2 degenerate poly-Bernoulli polynomials and present certain interesting graphical representations. The first five members of \(\beta _{\upsilon ,\lambda }^{(\kappa )}(\omega )\) are obtained and given as follows:

$$\begin{aligned} &\beta _{0,\lambda }^{(\kappa )}(\omega )=-\frac{1}{\log 3}, \\ &\beta _{1,\lambda }^{(\kappa )}(\omega )=\frac{1}{24 (\log 3)^{2}}+ \frac{1}{3 \log 3}-\frac{\omega }{\log 3}, \\ &\beta _{2,\lambda }^{(\kappa )}(\omega )=-\frac{2}{729 (\log 3)^{3}}- \frac{5}{72 (\log 3)^{2}}+ \frac{\omega }{12 (\log 3)^{2}}-\frac{43}{108 \log 3}+\frac{\omega }{\log 3}- \frac{\omega ^{2}}{\log 3}, \\ &\begin{aligned} \beta _{3,\lambda }^{(\kappa )}(\omega )&=\frac{8}{729 (\log 3)^{3}}- \frac{2 \omega }{243 (\log 3)^{3}}+ \frac{151}{864 (\log 3)^{2}}-\frac{\omega }{4 (\log 3)^{2}}+ \frac{\omega ^{2}}{8 (\log 3)^{2}}+ \frac{26}{27 \log 3} \\ &\quad {}-\frac{7 \omega }{4 \log 3}+\frac{2 \omega ^{2}}{\log 3}- \frac{\omega ^{3}}{\log 3}, \end{aligned} \\ &\begin{aligned}\beta _{4,\lambda }^{(\kappa )}(\omega )&=- \frac{313}{6561 (\log 3)^{3}}+ \frac{4 \omega }{81 (\log 3)^{3}}- \frac{4 \omega ^{2}}{243 (\log 3)^{3}}-\frac{805}{1296 (\log 3)^{2}}+ \frac{7 \omega }{8 (\log 3)^{2}} \\ &\quad {}-\frac{7 \omega ^{2}}{12 (\log 3)^{2}}+ \frac{\omega ^{3}}{6 (\log 3)^{2}}- \frac{63\text{,}667}{19\text{,}440 \log 3}+ \frac{31 \omega }{6 \log 3}\\ &\quad {}-\frac{89 \omega ^{2}}{18 \log 3}+\frac{10 \omega ^{3}}{3 \log 3}- \frac{\omega ^{4}}{\log 3}. \end{aligned} \end{aligned}$$

Next, we present some graphs showing the behavior of \(\beta _{\upsilon ,\lambda }^{(\kappa )}(\omega )\) for \(\kappa =4\) and \(\lambda =\frac{1}{3}\); these graphs are given in Fig. 1.

Figure 1
figure1

Graphs of \(\beta _{\upsilon ,\lambda }^{(\kappa )}(\omega )\)

The approximate solutions of \(\beta _{20,\lambda }^{(\kappa )}(\omega )=0\) when \(\kappa =4\) and \(\lambda =\frac{1}{3}\) are calculated and displayed in Fig. 2.

Figure 2
figure2

Zeros of \(\beta _{20,\lambda }^{(\kappa )}(\omega )\)

Further, we calculate the approximate zeros of \(\beta _{\upsilon ,\lambda }^{(\kappa )}(\omega )=0\) for \(\kappa =4\), \(\lambda =\frac{1}{3}\) and \(n=1, 2,\ldots, 20\), and show the stacking structure of these zeros in Fig. 3.

Figure 3
figure3

Stacking structure of zeros \(\beta _{\upsilon ,\lambda }^{(\kappa )}(\omega )\)

Conclusions

In this paper, we have studied and introduced degenerate versions of type 2 Bernoulli numbers and polynomials and derived some properties of these polynomials. We have given some relationships between higher-order Bernoulli polynomials, degenerate type 2 Bernoulli polynomials, degenerate central Bell polynomials, degenerate Stirling numbers of the first and second kind, degenerate central factorials numbers. Besides, we have introduced degenerate type 2 unipoly-Bernoulli polynomials by using degenerate unipoly polynomials and derived some identities of these polynomials. We have derived some relationship between degenerate type 2 Bernoulli polynomials and degenerate Daehee polynomials.

Availability of data and materials

Not applicable.

References

  1. 1.

    Carlitz, L.: A degenerate Staudt-Clausen theorem. Arch. Math. (Basel) 7, 28–33 (1956)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51–88 (1979)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Dolgy, D.V., Jang, G.-W., Kim, T.: A note on degenerate central factorial polynomials of the second kind. Adv. Stud. Contemp. Math. (Kyungshang) 29(1), 147–159 (2019)

    MATH  Google Scholar 

  4. 4.

    Dolgy, D.V., Khan, W.A.: A note on type two degenerate poly-Changhee polynomials of the second kind. Symmetry 13, 579, 1–12 (2021)

    Article  Google Scholar 

  5. 5.

    Hardy, G.H.: On the zeroes of certain classes of integral Taylor series. Part II.-On the integral function formula and other similar functions. Proc. Lond. Math. Soc. 2(2), 401–431 (1905)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Hardy, G.H.: On the zeroes certain classes of integral Taylor series. Part I.-On the integral function formula. Proc. Lond. Math. Soc. 2(2), 332–339 (1905)

    Article  Google Scholar 

  7. 7.

    Jang, G.-W., Kim, T.: A note on type 2 degenerate Euler and Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 29(1), 147–159 (2019)

    MATH  Google Scholar 

  8. 8.

    Jang, L.-C., Kim, D.S., Kim, H., Kim, T., Lee, H.: Study of degenerate poly-Bernoulli polynomials by λ-umbral calculus. Comput. Model. Eng. Sci. 2021 (2021). https://doi.org/10.32604/cmes.2021.016917

  9. 9.

    Kaneko, M.: Poly-Bernoulli numbers. J. Théor. Nr. Bordx. 9(1), 221–228 (1997)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Khan, W.A., Acikgoz, M., Duran, U.: Note on the type 2 degenerate multi-poly-Euler polynomials. Symmetry 12, 1691 (2020)

    Article  Google Scholar 

  11. 11.

    Khan, W.A., Ali, R., Alzobydi, K.A.H., Ahmed, N.: A new family of degenerate poly-Genocchi polynomials with its certain properties. J. Funct. Spaces 2021, Article ID 6660517 (2021)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Khan, W.A., Nisar, K.S., Duran, U., Acikgoz, M., Araci, S.: Multifarious implicit summation formulae of Hermite-based poly-Daehee polynomials. Appl. Math. Inf. Sci. 12(2), 305–310 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Khan, W.A., Sharma, S.K.: A new class of Hermite-based higher order central Fubini polynomials. Int. J. Appl. Comput. Math. 6(87), 1–14 (2020)

    MathSciNet  Google Scholar 

  14. 14.

    Kim, D.S., Kim, T.: A note on polyexponential and unipoly functions. Russ. J. Math. Phys. 26(1), 40–49 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Kim, D.S., Kim, T.: A note on a new type of degenerate Bernoulli numbers. Russ. J. Math. Phys. 27(2), 227–235 (2020)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Kim, D.S., Kwon, J., Dolgy, D.V., Kim, T.: On central Fubini polynomials associated with central factorial numbers of the second kind. Proc. Jangjeon Math. Soc. 20(3), 319–331 (2017)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Kim, T.: A note on degenerate Stirling numbers of the second kind. Proc. Jangjeon Math. Soc. 21(4), 589–598 (2018)

    MathSciNet  Google Scholar 

  18. 18.

    Kim, T., Jang, L.-C., Kim, D.S., Kim, H.-Y.: Some identities on type 2 degenerate Bernoulli polynomials of the second kind. Symmetry 12(4), Article ID 510 (2020)

    Article  Google Scholar 

  19. 19.

    Kim, T., Kim, D.S.: A note on type 2 Changhee and Daehee polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(3), 2783–2791 (2019)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Kim, T., Kim, D.S.: Degenerate central Bell numbers and polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 2507–2513 (2019)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Kim, T., Kim, D.S.: Degenerate polyexponential functions and degenerate Bell polynomials. J. Math. Anal. Appl. 487(2), 124017 (2020)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Kim, T., Kim, D.S.: Some identities of extended degenerate r-central Bell polynomials arising from umbral calculus. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 114(1), 1–19 (2020)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Kim, T., Kim, D.S.: A note on central Bell numbers and polynomials. Russ. J. Math. Phys. 27(1), 76–81 (2020)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Kim, T., Kim, D.S.: Degenerate zero-truncated Poisson random variables. Russ. J. Math. Phys. 28(1), 66–72 (2021)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Kim, T., Kim, D.S., Dolgy, D.V., Lee, S.-H., Kwon, J.: Some identities of the higher-order type 2 Bernoulli numbers and polynomials of the second kind. Comput. Model. Eng. Sci. 2021 (2021). https://doi.org/10.32604/cmes.2021.016532

  26. 26.

    Kim, T., Kim, D.S., Jang, L.-C., Kim, H.-Y.: On type 2 degenerate Bernoulli and Euler polynomials of complex variable. Adv. Differ. Equ. 2019, 490, 15 pp. (2019)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Kim, T., Kim, D.S., Jang, L.-C., Lee, H., Kim, H.: Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function. Adv. Differ. Equ. 2021, 175, 12 pp. (2021)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Kim, T., Kim, D.S., Kim, H.-Y., Kwon, J.: Some results on degenerate Daehee and Bernoulli numbers and polynomials. Adv. Differ. Equ. 2021, 311, 13 pp. (2020)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Kim, T., Kim, D.S., Kim, H.Y., Kwon, J.: Some identities of degenerate Bell polynomials. Mathematics 8(1), Article ID 40 (2020)

    Article  Google Scholar 

  30. 30.

    Kim, T., Kim, D.S., Kwon, J., Kim, H.-Y.: A note on degenerate Genocchi and poly-Genocchi numbers and polynomials. J. Inequal. Appl. 2020, 110, 13 pp. (2020)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Kim, T., Kim, D.S., Kwon, J., Lee, H.: Degenerate polyexponential functions and type 2 degenerate poly-Bernoulli numbers and polynomials. Adv. Differ. Equ. 2020, 168, 12 pp. (2020)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Kim, T., Kim, D.S., Kwon, J., Lee, H.: Representations of degenerate poly-Bernoulli polynomials. J. Inequal. Appl. 2021, 58, 12 pp. (2021)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Kim, T., Kim, D.S., Kwon, J., Lee, H., Park, S.-H.: Some properties of degenerate complete and partial Bell polynomials. Adv. Differ. Equ. 2021, 304, 12 pp. (2021)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Kim, T., Kim, D.S., Lee, H., Jang, L.-C.: A note on degenerate derangement polynomials and numbers. AIMS Math. 6(6), 6469–6481 (2021)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Ma, Y., Kim, D.S., Lee, H., Kim, H., Kim, T.: Reciprocity of poly-Dedkind-type DC sums involving poly-Euler functions. Adv. Differ. Equ. 2021, 30, 18 pp. (2021)

    Article  Google Scholar 

  36. 36.

    Ma, Y., Kim, T.: A note on negative λ-binomial distribution. Adv. Differ. Equ. 2021, 569, 7 pp. (2021)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Muhiuddin, G., Khan, W.A., Duran, U.: Two variable type 2 Fubini polynomials. Mathematics 9, 281, 1–13 (2021)

    Article  Google Scholar 

  38. 38.

    Muhiuddin, G., Khan, W.A., Duran, U., Al-Kadi, D.: Some identities of the multi-poly-Bernoulli polynomials of complex variable. J. Funct. Spaces 2021, Article ID 7172054 (2021). https://doi.org/10.1155/2021/7172054

    MathSciNet  Article  MATH  Google Scholar 

  39. 39.

    Muhiuddin, G., Khan, W.A., Muhyi, A., Al-Kadi, D.: Some results on type 2 degenerate poly-Fubini polynomials and numbers. Comput. Model. Eng. Sci. 2021, 23 pp. (2021). https://doi.org/10.32604/cmes.2021.016546

  40. 40.

    Ryoo, C.S., Khan, W.A.: On two bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials. Mathematics 8, 417 (2020). https://doi.org/10.3390/math8030417

    Article  Google Scholar 

  41. 41.

    Sharma, S.K., Khan, W.A., Araci, S., Ahmed, S.S.: New type of degenerate Daehee polynomials of the second kind. Adv. Differ. Equ. 2020, 428, 14 pp. (2020)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Sharma, S.K., Khan, W.A., Araci, S., Ahmed, S.S.: New construction of type 2 degenerate central Fubini polynomials with their certain properties. Adv. Differ. Equ. 2020, 587, 11 pp. (2020)

    MathSciNet  Article  Google Scholar 

  43. 43.

    Sharma, S.K., Khan, W.A., Ryoo, C.S.: A parametric kind of the degenerate Fubini numbers and polynomials. Mathematics 8, 405 (2020). https://doi.org/10.3390/math8030405

    Article  Google Scholar 

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Acknowledgements

The authors are grateful to the anonymous referees for careful checking of the details and for helpful comments that improved this paper.

Funding

This work was supported by the Taif University Researchers Supporting Project (TURSP-2020/246), Taif University, Taif, Saudi Arabia.

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All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.

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Correspondence to Waseem A. Khan.

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Khan, W.A., Muhiuddin, G., Muhyi, A. et al. Analytical properties of type 2 degenerate poly-Bernoulli polynomials associated with their applications. Adv Differ Equ 2021, 420 (2021). https://doi.org/10.1186/s13662-021-03575-7

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MSC

  • 11B73
  • 11B83
  • 05A19

Keywords

  • Modified degenerate polyexponential functions
  • Type 2 degenerate Bernoulli polynomials
  • Type 2 degenerate central Bell polynomials
  • Unipoly functions