New Families of Certain Special Polynomials: A Kaniadakis Calculus Viewpoint

Abstract

In this paper, we introduce a new family of Stirling polynomials of the second kind, Bell polynomials, bivariate Bell polynomials, Bernoulli polynomials of higher order, and Euler polynomials of higher order arising from the Kaniadakis calculus viewpoint. We refer to each of them as $\kappa$-polynomials. Through the defined concepts of Kaniadakis calculus, we derive explicit formulas, summation formulas, and addition formulas for the polynomials discussed in the present paper. We also present the Volkenborn integral and the fermionic p-adic integral representations in terms of the $\kappa$-Stirling polynomials of the second kind, bivariate $\kappa$-Bell polynomials, $\kappa$-Bernoulli polynomials of higher order, and $\kappa$-Euler polynomials of higher order. We establish some formulae, including old and new polynomials. Finally, we investigate determinantal representations for the $\kappa$-Euler polynomials and the $\kappa$-Bernoulli polynomials.

1. Introduction

The fundamental elements of the $\kappa$-deformed framework, introduced by Giorgio Kaniadakis, are presented in [1]. The Kaniadakis non-Gaussian statistics are characterized by the $\kappa$-entropy, which naturally arises within the framework of the kinetic interaction principle underlying nonlinear kinetics in particle systems. This principle determines the form of the Fokker–Planck equation describing the kinetic evolution of such systems and fixes the corresponding generalized entropy. The structure of the resulting $\kappa$-deformed statistical mechanics exhibits a remarkable analogy with special relativity, suggesting its relevance for the self-consistent formulation of relativistic statistical theory [2]. Applications include, for example, the relativistic distribution of cosmic ray fluxes [2], the formation of quark–gluon plasma, the relativistic nuclear equation of state, relativistic gases in electromagnetic fields, and the relaxation of relativistic plasmas under wave–particle interactions. Beyond these fundamental aspects, $\kappa$-deformed statistics have also been successfully applied in diverse areas of physics. Illustrative examples include studies of sensitivity to initial conditions and entropy production in the logistic map, nonlinear kinetics and the H-theorem from a generalized molecular chaos hypothesis, as well as applications in socio-economic and astrophysical contexts, such as modeling personal income distributions in the United Kingdom, Italy, and Germany, and describing the distribution of stellar rotational velocities for various stellar classes. For further details, see [3] and the references cited therein.

Mathematically, the statistics introduced by Kaniadakis are based on the $\kappa$-deformed exponential ${exp}_{\kappa }\left(t\right)$ and logarithm ${ln}_{\kappa }\left(t\right)$ functions [4], defined by

$${exp}_{\kappa }\left(t\right)={\left[\sqrt{1+{\kappa }^2{t}^2}+\kappa t\right]}^{\frac{1}{\kappa }}=e^{\frac{1}{\kappa }arcsinh\left(\kappa t\right)}$$

and

$${ln}_{\kappa }\left(t\right)=\frac{t^{\kappa }-t^{-\kappa }}{2\kappa }=\frac{1}{\kappa }sinh(\kappa lnt).$$

Based on these functions, Kaniadakis Calculus has many applications in fractal systems, dynamical systems, fracture propagation, quantum hadrodynamics, quark–gluon plasmas, stellar distributions in astrophysics, particle kinetics in the presence of temperature gradients, taxation redistribution model, equity options, finance, economic systems logit models, economic systems income, information theory, networks, game theory, error theory, random matrix theory, field theories, particles in an external conservative force field, interacting particle systems, nonlinear kinetics, nonlinear diffusion, relaxation in relativistic plasmas, relativistic and classical plasma physics in external em fields, mathematical aspects, geometrical aspects, quantum statistical mechanics, nonequilibrium thermodynamics, Gibbs theorem, thermodynamics, Legendre structures, Lesche stability, thermodynamics stability, molecular chaos hypothesis, and so on (cf. [5,6,7] and see also the references cited therein). However, in this work, we focus on introducing new families of special numbers and polynomials using the mathematical tools of Kaniadakis calculus, and derive novel formulae and relations.

The outline of this paper is as follows. Section 1 is the introduction, which will include literature review regarding the theory of Kaniadakis’ $\kappa$-deformed framework and its applications. Section 2 will mainly deal with some definitions, formulas and results belonging to Kaniadakis calculus. Kaniadakis Stirling polynomials of the second kind are defined, and some of their properties will be discussed in Section 3. Also, $\kappa$-Bell polynomials will be introduced, and some properties will be analyzed in Section 4. In Section 5, Kaniadakis Bernoulli and Euler polynomials will be considered and some formulas for them will be examined. In Section 6, several correlations will be mentioned. Section 7 will provide Volkenborn integral and fermionic p-adic integral representations in terms of the $\kappa$-Stirling polynomials of the second kind, bivariate $\kappa$-Bell polynomials, $\kappa$-Bernoulli polynomials of order $\alpha$, and $\kappa$-Euler polynomials of order $\alpha$, and will also present diverse formulas including old and new polynomials. Furthermore, determinantal representations for the $\kappa$-Euler polynomials and the $\kappa$-Bernoulli polynomials will be investigated. Finally, the conclusions and findings gained in this study will be presented.

2. Review of Kaniadakis Calculus

Here, we review Kaniadakis calculus (which can also be called $\kappa$-calculus) (cf. [4]); see the references cited therein).

Let $t,u\in \mathbb{R}$ and $-1<\kappa <1$. The composition law $\stackrel{\kappa }{\oplus }$ defined through

$$t\stackrel{\kappa }{\oplus }u=t\sqrt{1+{\kappa }^2{u}^2}+u\sqrt{1+{\kappa }^2{t}^2},$$

(1)

is a generalized sum called $\kappa$-sum and the algebraic structure $(\mathbb{R},\stackrel{\kappa }{\oplus })$ forms an abelian group. The $\kappa$-difference $\stackrel{\kappa }{\ominus }$ is defined as $t\stackrel{\kappa }{\ominus }u=t\stackrel{\kappa }{\oplus }(-u)$. The $\kappa$-sum is a one-parameter continuous deformation of the ordinary sum, which recovers in the classical limit $\kappa \to 0$, i.e., $t\stackrel{0}{\oplus }u=t+u$.

Let $t,u\in \mathbb{R}$ and $-1<\kappa <1$. The composition law $\stackrel{\kappa }{\otimes }$ defined through

$$t\stackrel{\kappa }{\otimes }u=\frac{1}{\kappa }sinh\left(\frac{1}{\kappa }\mathrm{arcsinh}\left(\kappa t\right)+\mathrm{arcsinh}\left(\kappa u\right)\right),$$

(2)

is a generalized product, called $\kappa$-product and the algebraic structure $(\mathbb{R},\stackrel{\kappa }{\otimes })$ forms an abelian group. The $\kappa$-product reduces to the ordinary product as $\kappa \to 0$, i.e., $t\stackrel{0}{\otimes }u=tu$. The $\kappa$-division $\stackrel{\kappa }{\oslash }$ is defined through $t\stackrel{\kappa }{\oslash }u=t\stackrel{\kappa }{\otimes }\overline{u}$, where

$$\overline{u}={\kappa }^{-1}sinh\left(\frac{{\kappa }^2}{arcsinh\left(\kappa u\right)}\right).$$

Let

$$\left\{t\right\}:=\frac{1}{\kappa }\mathrm{arcsinh}\kappa t,$$

it follows that the cyclic functions are defined in the interval $-1/\kappa \le t\le 1/\kappa$. The function

$$\left[t\right]:=\frac{1}{\kappa }sinh\kappa t,$$

is defined as the inverse of $\left\{t\right\}$, i.e., $\left[\right\{t\left\}\right]=\left\{\right[t\left]\right\}=t$. The $\kappa$-sum $\stackrel{\kappa }{\oplus }$ and $\kappa$-product $\stackrel{\kappa }{\otimes }$ given in (1) and (2) are isomorphic operations for the ordinary sum and product, respectively, i.e.,

$$\left\{t\stackrel{\kappa }{\oplus }u\right\}=\left\{t\right\}+\left\{u\right\},$$

and

$$\left\{t\stackrel{\kappa }{\otimes }u\right\}=\left\{t\right\}·\left\{u\right\}.$$

Let $t\in \mathbb{R}$ and n be an arbitrary nonnegative integer. It holds

$$\underset{n\mathrm{times}}{\underbrace{t\stackrel{\kappa }{\oplus }t\stackrel{\kappa }{\oplus }\dots \stackrel{\kappa }{\oplus }t}}=\left[n\right]\stackrel{\kappa }{\otimes }t.$$

The $\kappa$-exponential function is defined by

$${exp}_{\kappa }\left(t\right)={\left(\sqrt{1+{\kappa }^2{t}^2}+\kappa t\right)}^{\frac{1}{\kappa }}=e^{\frac{1}{\kappa }\mathrm{arcsinh}\left(\kappa t\right)}=\sum _{n=0}^{\infty }{\kappa }^{-n}\frac{{\left(\mathrm{arcsinh}\left(\kappa t\right)\right)}^n}{n!},$$

(3)

then, we have

$$\frac{d{exp}_{\kappa }\left(t\right)}{d_{\kappa }t}={exp}_{\kappa }\left(t\right).$$

It is clear from (3) that

$$\underset{\kappa \to 0}{lim}{exp}_{\kappa }\left(t\right)=exp\left(t\right)\mathrm{and}{exp}_{-\kappa }\left(t\right)={exp}_{\kappa }\left(t\right).$$

Like the ordinary exponential, ${exp}_{\kappa }\left(t\right)$ has the properties

${exp}_{\kappa }\left(t\right)\in C^{\infty }\left(\mathbb{R}\right),$	$\frac{d}{dt}{exp}_{\kappa }\left(t\right)>0,$	${exp}_{\kappa }(-\infty )=0^{+},$
${exp}_{\kappa }\left(0\right)=1,$	${exp}_{\kappa }(+\infty )=+\infty ,$	${exp}_{\kappa }\left(t\right){exp}_{\kappa }(-t)=1,$

and

$${exp}_{\kappa }\left(t\right){exp}_{\kappa }\left(u\right)={exp}_{\kappa }\left(t\stackrel{\kappa }{\oplus }u\right).$$

(4)

Furthermore, ${exp}_{\kappa }\left(t\right)$ has the property

$${\left({exp}_{\kappa }\left(t\right)\right)}^r={exp}_{\kappa /r}\left(rt\right),$$

with $r\in \mathbb{R}$, which in the limit $\kappa \to 0$ reproduces one well-known property of the ordinary exponential function, as follows:

$${\left(e^t\right)}^r=e^{rt}.$$

The Taylor expansion of ${exp}_{\kappa }\left(t\right)$ given in [4] can be also written in the following form

$${exp}_{\kappa }\left(t\right)=\sum _{n=0}^{\infty }\frac{t^n}{n{!}_{\kappa }};{\kappa }^2{t}^2<1,$$

where the symbol $n{!}_{\kappa }$, representing the $\kappa$-generalization of the ordinary factorial $n!$, recovered for $\kappa =0$, is given by

$$n{!}_{\kappa }=\frac{n!}{a_n\left(\kappa \right)},$$

and the polynomials $a_n\left(\kappa \right)$ are defined as

$$a_n\left(\kappa \right)=\prod _{i=1}^{n-1}\left(1-(2i-n)\kappa \right)\mathrm{for}n>1,$$

with initial values $a_0\left(\kappa \right)=1$ and $a_1\left(\kappa \right)=1$. The first nine $a_n\left(\kappa \right)$ polynomials read as

$$\begin{array}{cc}\hfill a_0\left(\kappa \right)& =a_1\left(\kappa \right)=a_2\left(\kappa \right)=1,\hfill \\ \hfill a_3\left(\kappa \right)& =1-{\kappa }^2,\hfill \\ \hfill a_4\left(\kappa \right)& =1-4{\kappa }^2,\hfill \\ \hfill a_5\left(\kappa \right)& =(1-{\kappa }^2)(1-9{\kappa }^2),\hfill \\ \hfill a_6\left(\kappa \right)& =(1-4{\kappa }^2)(1-16{\kappa }^2),\hfill \\ \hfill a_7\left(\kappa \right)& =(1-{\kappa }^2)(1-9{\kappa }^2)(1-25{\kappa }^2),\hfill \\ \hfill a_8\left(\kappa \right)& =(1-4{\kappa }^2)(1-16{\kappa }^2)(1-36{\kappa }^2).\hfill \end{array}$$

The Taylor series expansion of $\kappa$-exponential function can also be expressed in terms of the polynomials $a_n\left(\kappa \right)$ as follows:

$${exp}_{\kappa }^x\left(t\right):={\left(\sqrt{1+{\kappa }^2{t}^2}+\kappa t\right)}^{\frac{x}{\kappa }}=\sum _{n=0}^{\infty }{a}_n\left(\frac{\kappa }{x}\right)x^n\frac{t^n}{n!}.$$

(5)

In the $\kappa$-calculus, the $\kappa$-logarithm is defined as the inverse function of ${exp}_{\kappa }\left(t\right)$, namely

$${ln}_{\kappa }\left({exp}_{\kappa }t\right)={exp}_{\kappa }\left({ln}_{\kappa }t\right)=t,$$

and is given by

$${ln}_{\kappa }\left(t\right)=[lnt]=\frac{1}{\kappa }sinh(\kappa lnt),$$

or more properly

$${ln}_{\kappa }\left(t\right)=\frac{t^{\kappa }-t^{-\kappa }}{2\kappa }.$$

(6)

It results that

$${ln}_0\left(t\right)\equiv \underset{\kappa \to 0}{lim}{ln}_{\kappa }\left(t\right)=ln\left(t\right)\mathrm{and}{ln}_{-\kappa }\left(t\right)={ln}_{\kappa }\left(t\right).$$

The function ${ln}_{\kappa }\left(t\right)$, just as the ordinary logarithm, has the following properties:

$$\begin{array}{cccccc}\hfill {ln}_{\kappa }\left(t\right)& \in C^{\infty }\left({\mathbb{R}}^{+}\right),\hfill & \hfill {ln}_{\kappa }\left(0^{+}\right)& =-\infty ,\hfill & \hfill {ln}_{\kappa }\left(1\right)& =0,\hfill \\ \hfill \frac{d}{dt}{ln}_{\kappa }\left(t\right)& >0,\hfill & \hfill {ln}_{\kappa }(+\infty )& =+\infty ,\hfill & \hfill {ln}_{\kappa }(1/t)& =-{ln}_{\kappa }\left(t\right).\hfill \end{array}$$

Furthermore, ${ln}_{\kappa }\left(t\right)$ has the following properties

$${ln}_{\kappa }\left(t^r\right)=r{ln}_{r\kappa }\left(t\right)\mathrm{and}{ln}_{\kappa }(t.u)={ln}_{\kappa }\left(t\right)\stackrel{\kappa }{\oplus }{ln}_{\kappa }\left(u\right),$$

(7)

with $r\in \mathbb{R}$.

Also, we have

$${ln}_{\kappa }\left(t\right)=lnt\prod _{i=1}^{\infty }\left(1+\frac{{\left(\kappa lnt\right)}^2}{i^2{\pi }^2}\right).$$

The Taylor expansion of ${ln}_{\kappa }\left(1+t\right)$ about $t=0$ is given by

$${ln}_{\kappa }\left(1+t\right)=t-\frac{t^2}{2}+\left(1+\frac{{\kappa }^2}{2}\right)\frac{t^3}{3}-\cdots =\sum _{n=0}^{\infty }{b}_n\left(\kappa \right){\left(-1\right)}^{n-1}\frac{t^n}{n},$$

(8)

with $b_1\left(\kappa \right)=1,$ where

$$b_n\left(\kappa \right)=\left\{\begin{array}{cc}1& \mathrm{if}n=1,\\ \frac{1}{2}\left(1-\kappa \right)\left(1-\frac{\kappa }{2}\right)\cdots \left(1-\frac{\kappa }{n-1}\right)+\frac{1}{2}\left(1+\kappa \right)\left(1+\frac{\kappa }{2}\right)\cdots \left(1+\frac{\kappa }{n-1}\right)& \mathrm{if}n>1,\end{array}\right.$$

and $b_n\left(0\right)=1.$ Also note that $b_n\left(\kappa \right)=b_n\left(-\kappa \right)$.

We observe that

$${ln}_{\kappa }\left(t\right)=\frac{t^{\kappa }-t^{-\kappa }}{2\kappa }=\frac{e^{\kappa lnt}-e^{-\kappa lnt}}{2\kappa }=\frac{sinh\left(\kappa lnt\right)}{\kappa }=\sum _{n=0}^{\infty }{\kappa }^{2n}\frac{{\left(lnt\right)}^{2n+1}}{\left(2n+1\right)!}.$$

The term Kaniadakis statistics (also known as $\kappa$-statistics) refers to a generalization of Boltzmann–Gibbs statistical mechanics based on a relativistic generalization of the classical Boltzmann–Gibbs–Shannon entropy (commonly referred to as Kaniadakis entropy or $\kappa$-entropy), cf. [2,6]. Also, the $\kappa$-distribution is one of the most viable candidates for explaining complex physical, natural, or artificial systems involving power-law-tailed statistical distributions, cf. [6]. Kaniadakis statistics have been adopted successfully in the description of a variety of systems in the fields of cosmology, astrophysics, condensed matter, quantum physics, seismology, genomics, economics, epidemiology, and many others, cf. [5,6,7]; see the references cited therein for further details.

3. $\mathsf{\kappa }$-Stirling Numbers and Polynomials of the Second Kind

In this part, we aim to define $\kappa$-extension of the Stirling numbers and polynomials of the second kind and also to investigate their numerous formulas and relations.

The Stirling numbers $S_2\left(n,j\right)$ and polynomials $S_2\left(n,j:x\right)$ of the second kind are provided as follows (cf. [8,9]):

$$\sum _{n=0}^{\infty }{S}_2\left(n,j\right)\frac{t^n}{n!}=\frac{{\left(e^t-1\right)}^j}{j!}\mathrm{and}\sum _{n=0}^{\infty }{S}_2\left(n,j:x\right)\frac{t^n}{n!}=\frac{{\left(e^t-1\right)}^j}{j!}{e}^{tx}.$$

(9)

The Stirling numbers of the second kind, in combinatorics, count the number of ways in which n distinguishable objects can be partitioned into j indistinguishable subsets when each subset has to contain at least one object, cf. [8,9].

By replacing the ordinary exponential function $e^t$ with the deformed exponential function ${exp}_{\kappa }\left(t\right)$, we introduce new families of Stirling numbers and polynomials of the second kind, referred to as the $\kappa$-Stirling numbers and $\kappa$-Stirling polynomials of the second kind, defined as follows.

Definition 1.

The κ-Stirling numbers $S_{2,\kappa }\left(n,j\right)$ and polynomials $S_{2,\kappa }\left(n,j:x\right)$ of the second kind are introduced as follows:

$$\sum _{n=j}^{\infty }{S}_{2,\kappa }\left(n,j\right)\frac{t^n}{n!}=\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}and\sum _{n=j}^{\infty }{S}_{2,\kappa }\left(n,j:x\right)\frac{t^n}{n!}=\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}{exp}_{\kappa }\left(xt\right).$$

(10)

Remark 1.

In the limit $\kappa \to 0$, the κ-Stirling numbers and polynomials reduce to the classical Stirling numbers and polynomials of the second kind. More precisely,

$$\underset{\kappa \to 0}{lim}{S}_{2,\kappa }(n,j)=S_2(n,j),\underset{\kappa \to 0}{lim}{S}_{2,\kappa }(n,j:x)=S_2(n,j:x),$$

where $S_2(n,j)$ and $S_2(n,j:x)$ denote the ordinary Stirling numbers and polynomials of the second kind, as defined in (9).

It can be readily seen from (10) that $S_{2,\kappa }\left(n,j:0\right)=S_{2,\kappa }\left(n,j\right)$.

Remark 2.

We investigate from (10) that

$$\begin{array}{c}{S}_{2,\kappa }\left(n,j\right)=0forj<norforn<0orforj<0;\\ S_{2,\kappa }\left(n,n\right)=1for0\le n,\end{array}$$

and

$$S_{2,\kappa }\left(n,0\right)=S_{2,\kappa }\left(0,n\right)=0\mathrm{for}n>0.$$

Now, we give explicit formula for $\kappa$-Stirling numbers of the second kind by the following theorem.

Theorem 1.

The following explicit formula

$$S_{2,\kappa }\left(n,j\right)=\frac{1}{j!}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right){\left(-1\right)}^{j-l}{a}_n\left(\kappa /l\right)l^n$$

(11)

is valid for $j,n\in {\mathbb{N}}_0$ with $j\le n$.

Proof. 

We observe from (5) and (10) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{S}_{2,\kappa }\left(n,j\right)\frac{t^n}{n!}& =& \frac{1}{j!}{\left({exp}_{\kappa }\left(t\right)-1\right)}^j\hfill \\ & =& \frac{1}{j!}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right){\left(-1\right)}^{j-l}{exp}_{\kappa }^l\left(t\right)\hfill \\ & =& \frac{1}{j!}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right){\left(-1\right)}^{j-l}\sum _{n=0}^{\infty }{a}_n\left(\frac{\kappa }{l}\right)l^n\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\frac{1}{j!}\sum _{l=0}^j\left(\genfrac{}{}{0pt}{}{j}{l}\right){\left(-1\right)}^{j-l}{a}_n\left(\kappa /l\right)l^n\frac{t^n}{n!}.\hfill \end{array}$$

Hence, the computations above refer to the claimed result (11). □

We readily see from (11) that

$$S_{2,\kappa }\left(n,1\right)=a_n\left(\kappa \right)\mathrm{for}n\ge 0.$$

Also, utilizing explicit formulas of $\kappa$-Stirling numbers of the second kind as shown in (11), we compute the first few $\kappa$-Stirling numbers $S_{2,\kappa }\left(n,j\right)$ of the second kind in

Table 1. The list of $\kappa$-Stirling numbers of the second kind.

$\mathit{j}/\mathit{n}$	1	2	3	4	5	6
1	1	1	$1-{\kappa }^2$	$1-4{\kappa }^2$	$1-10{\kappa }^2+9{\kappa }^4$	$1-20{\kappa }^2+64{\kappa }^4$
2	0	1	3	$7-{\kappa }^2$	$15-30{\kappa }^2$	$31-140{\kappa }^2+64{\kappa }^4$
3	0	0	1	6	25	$90-120{\kappa }^2$
4	0	0	0	1	10	$65-20{\kappa }^2$
5	0	0	0	0	1	15
6	0	0	0	0	0	1

:

The following theorem states that $\kappa$-Stirling polynomials of the second kind can be expressed in terms of $\kappa$-Stirling numbers of the second kind.

Theorem 2.

The following relation

$$S_{2,\kappa }\left(n,j:x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)S_{2,\kappa }\left(n-l,j\right)a_l\left(\kappa \right)x^l$$

(12)

holds for $j,n\in {\mathbb{N}}_0$ with $j\le n$.

Proof. 

We observe from (5) and (10) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{S}_{2,\kappa }\left(n,j:x\right)\frac{t^n}{n!}& =& \frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}{exp}_{\kappa }\left(xt\right)\hfill \\ & =& \sum _{n=0}^{\infty }{S}_{2,\kappa }\left(n,j\right)\frac{t^n}{n!}\sum _{n=0}^{\infty }{a}_n\left(\kappa \right)x^n\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)S_{2,\kappa }\left(n-l,j\right)a_l\left(\kappa \right)x^l\frac{t^n}{n!}.\hfill \end{array}$$

Hence, the computations above indicate the claimed result (12). □

Under $\kappa$-summation property, we give the following addition formulae.

Theorem 3.

The following identity holds true:

$$S_{2,\kappa }\left(n,j:x\stackrel{\kappa }{\oplus }y\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)S_{2,\kappa }\left(l,j:x\right)a_{n-l}\left(\kappa \right)y^{n-l}$$

(13)

holds for $j,n\in {\mathbb{N}}_0$ with $j\le n$.

Proof. 

We acquire from (10) that

$$\begin{array}{ccc}\hfill \sum _{n=j}^{\infty }{S}_{2,\kappa }\left(n,j:x\stackrel{\kappa }{\oplus }y\right)\frac{t^n}{n!}& =& \frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}{exp}_{\kappa }\left(\left(x\stackrel{\kappa }{\oplus }y\right)t\right)\hfill \\ & =& \frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}{exp}_{\kappa }\left(xt\right){exp}_{\kappa }\left(yt\right)\hfill \\ & =& \sum _{n=0}^{\infty }{S}_{2,\kappa }\left(n,j:x\right)\frac{t^n}{n!}\sum _{n=0}^{\infty }{a}_n\left(\kappa \right)y^n\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)S_{2,\kappa }\left(l,j:x\right)a_{n-l}\left(\kappa \right)y^{n-l}\frac{t^n}{n!},\hfill \end{array}$$

which means the alleged equality (13). □

The $\kappa$-Stirling polynomials of the second kind can be written as the sums for the products of $\kappa$-Stirling numbers and polynomials of the second kind by the following theorem.

Theorem 4.

The following relation

$$S_{2,\kappa }\left(n,j+l:x\right)=\frac{j!l!}{\left(j+l\right)!}\sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)S_{2,\kappa }\left(n-i,j:x\right)S_{2,\kappa }\left(i,l\right)$$

(14)

is valid for $l,n,j\in {\mathbb{N}}_0$ with $n\ge l+j$.

Proof. 

Given (10), we have

$$\begin{array}{c}\sum _{n=0}^{\infty }{S}_{2,\kappa }\left(n,j+l:x\right)\frac{t^n}{n!}=\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^{j+l}}{\left(j+l\right)!}{exp}_{\kappa }\left(xt\right)\\ =\frac{j!l!}{\left(j+l\right)!}\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}{exp}_{\kappa }\left(xt\right)\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^l}{l!}\\ =\frac{j!l!}{\left(j+l\right)!}\sum _{n=j}^{\infty }{S}_{2,\kappa }\left(n,j:x\right)\frac{t^n}{n!}\sum _{n=l}^{\infty }{S}_{2,\kappa }\left(n,l\right)\frac{t^n}{n!}\\ =\frac{j!l!}{\left(j+l\right)!}\sum _{n=0}^{\infty }\sum _{i=0}^n\left(\genfrac{}{}{0pt}{}{n}{i}\right)S_{2,\kappa }\left(n-i,j:x\right)S_{2,\kappa }\left(i,l\right)\frac{t^n}{n!},\end{array}$$

which means the alleged consequence (14). □

The $\kappa$-Stirling polynomials of the second kind at the value $x=1$ can be stated as a recurrence relation by the following theorem.

Theorem 5.

The following relation

$$S_{2,\kappa }\left(n,j:1\right)=\left(j+1\right)S_{2,\kappa }\left(n,j+1\right)+S_{2,\kappa }\left(n,j\right)$$

(15)

holds for $n,j\in {\mathbb{N}}_0$ with $j\le n$.

Proof. 

We compute from (10) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{S}_{2,\kappa }\left(n,j:1\right)\frac{t^n}{n!}& =& \frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}\left({exp}_{\kappa }\left(t\right)-1+1\right)\hfill \\ & =& \left(j+1\right)\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^{j+1}}{\left(j+1\right)!}+\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}\hfill \\ & =& \left(j+1\right)\sum _{n=j+1}^{\infty }{S}_{2,\kappa }\left(n,j+1\right)\frac{t^n}{n!}+\sum _{n=j}^{\infty }{S}_{2,\kappa }\left(n,j\right)\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\left((j+1)S_{2,\kappa }\left(n,j+1\right)+S_{2,\kappa }\left(n,j\right)\right)\frac{t^n}{n!},\hfill \end{array}$$

which presents the desired result (15). □

From (12) and (15), we procure the following corollary.

Corollary 1.

We have

$$S_{2,\kappa }\left(n,1+j\right)=\frac{1}{1+j}\sum _{l=1}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)S_{2,\kappa }\left(n-l,j\right)a_l\left(\kappa \right).$$

(16)

4. Bivariate $\mathsf{\kappa }$-Bell Polynomials

In this part, we aim to define $\kappa$-extension of the bivariate Bell polynomials, and also to investigate a number of their formulas and relations.

The bivariate (or say, two-variable) Bell polynomials, ${\varphi }_n\left(x,y\right)$, are provided by (cf. [10,11,12]):

$$e^{x\left(e^t-1\right)+yt}=\sum _{n=0}^{\infty }{\varphi }_n\left(x,y\right)\frac{t^n}{n!}.$$

(17)

The usual Bell polynomials (or say, Touchard polynomials) are obtained by choosing $y=0$ in (17), namely ${\varphi }_n\left(x\right):={\varphi }_n\left(x,0\right)$, and are considered as follows (cf. [13,14,15,16]):

$$e^{x\left(e^t-1\right)}=\sum _{n=0}^{\infty }{\varphi }_n\left(x\right)\frac{t^n}{n!}.$$

(18)

The usual Bell numbers are obtained by choosing $x=1$ in (18), namely ${\varphi }_n:={\varphi }_n\left(1\right)$ and are provided as follows:

$$e^{\left(e^t-1\right)}=\sum _{n=0}^{\infty }{\varphi }_n\frac{t^n}{n!}.$$

These numbers, considered by Bell [17], emerge as a standard mathematical tool and arise in combinatorial analysis. Since the initial examination of the Bell numbers, these numbers have been deeply developed and worked on by many scientists [10,11,12,13,14,15,16,17].

The numbers $S_2\left(n,m\right)$ and the polynomials ${\varphi }_n\left(x\right)$ fulfill the following well-known correlation (cf. [8,9,12])

$${\varphi }_n\left(x\right)=\sum _{m=0}^n{S}_2\left(n,m\right)x^m.$$

(19)

Now, we aim to define $\kappa$-extension of the bivariate Bell polynomials and to derive diverse formulas and relations.

Definition 2.

The bivariate κ-Bell polynomials are introduced as follows:

$$\sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x,y\right)\frac{t^n}{n!}=e^{x\left({exp}_{\kappa }\left(t\right)-1\right)}{exp}_{\kappa }\left(yt\right).$$

(20)

Remark 3.

By taking the limit as $\kappa \to 0$, the function ${\varphi }_{n,\kappa }(x,y)$ reduces to its classical version:

$$\underset{\kappa \to 0}{lim}{\varphi }_{n,\kappa }(x,y)={\varphi }_n(x,y).$$

Remark 4.

Choosing $y=0$ in (20), the polynomials ${\varphi }_{n,\kappa }\left(x,y\right)$ become the κ-Bell polynomials ${\varphi }_{n,\kappa }\left(x\right)$, which is a new family of polynomials, as follows:

$$\sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x\right)\frac{t^n}{n!}=e^{x\left({exp}_{\kappa }\left(t\right)-1\right)}.$$

(21)

Remark 5.

Choosing $x=1$ in (20), the polynomials ${\varphi }_{n,\kappa }\left(x,y\right)$ become another κ-Bell polynomial ${\varphi }_{n,\kappa }\left(x\right)$, which represents a new family of polynomials, as follows:

$$\sum _{n=0}^{\infty }{\varphi }_n^{\kappa }\left(y\right)\frac{t^n}{n!}=e^{\left({exp}_{\kappa }\left(t\right)-1\right)}{exp}_{\kappa }\left(yt\right).$$

(22)

Remark 6.

Replacing $x=1=y+1$ in (20), the polynomials ${\varphi }_{n,\kappa }\left(x,y\right)$ become the κ-Bell numbers ${\varphi }_{n,\kappa }$, which are a new family of numbers, as follows:

$$\sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\frac{t^n}{n!}=e^{\left({exp}_{\kappa }\left(t\right)-1\right)}.$$

(23)

The bivariate $\kappa$-Bell polynomials can be formulated as stated in the following theorem.

Theorem 6.

The following summation formula

$${\varphi }_{n,\kappa }\left(x,y\right)=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x\right)a_j\left(\kappa \right)y^j$$

(24)

holds for $n\in {\mathbb{N}}_0$.

Proof. 

It is readily seen from (5), (20) and (21) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x,y\right)\frac{t^n}{n!}& =& e^{x\left({exp}_{\kappa }\left(t\right)-1\right)}{exp}_{\kappa }\left(yt\right)\hfill \\ & =& \sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x\right)\frac{t^n}{n!}\sum _{n=0}^{\infty }{a}_n\left(\kappa \right)y^n\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x\right)a_j\left(\kappa \right)y^j\frac{t^n}{n!},\hfill \end{array}$$

which gives the alleged result (24). □

Some special cases of (24) are as follows.

Corollary 2.

When $y=1$ in (24), we have

$${\varphi }_{n,\kappa }\left(x,1\right)=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x\right)a_j\left(\kappa \right).$$

Corollary 3.

When $x=1$ in (24), we have

$${\varphi }_n^{\kappa }\left(y\right)=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }{a}_j\left(\kappa \right)y^j.$$

The following theorem provides the addition formula for the bivariate $\kappa$-Bell polynomials with respect to the variables x and y under both classical and $\kappa$-summation.

Theorem 7.

The following identity

$${\varphi }_{n,\kappa }\left(x_1+x_2,y_1\stackrel{\kappa }{\oplus }{y}_2\right)=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x_1,y_1\right){\varphi }_{j,\kappa }\left(x_2,y_2\right)$$

(25)

holds for $n\in {\mathbb{N}}_0$.

Proof. 

It is readily seen from (20) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x_1+x_2,y_1\stackrel{\kappa }{\oplus }{y}_2\right)\frac{t^n}{n!}& =& e^{\left(x_1+x_2\right)\left({exp}_{\kappa }\left(t\right)-1\right)}{exp}_{\kappa }\left(\left(y_1\stackrel{\kappa }{\oplus }{y}_2\right)t\right)\hfill \\ & =& e^{x_1\left({exp}_{\kappa }\left(t\right)-1\right)}{exp}_{\kappa }\left(y_{1}t\right)e^{x_2\left({exp}_{\kappa }\left(t\right)-1\right)}{exp}_{\kappa }\left(y_{2}t\right)\hfill \\ & =& \sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x_1,y_1\right)\frac{t^n}{n!}\sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x_2,y_2\right)\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x_1,y_1\right){\varphi }_{j,\kappa }\left(x_2,y_2\right)\frac{t^n}{n!},\hfill \end{array}$$

which gives the alleged result (25). □

Two direct outcomes of (25) are analyzed as follows.

Corollary 4.

For $n,m\in {\mathbb{N}}_0$, we have

$${\varphi }_{n,\kappa }\left(x_1+x_2,y\right)=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x_1,y\right){\varphi }_{j,\kappa }\left(x_2\right)$$

and

$${\varphi }_{n,\kappa }\left(x,y_1\stackrel{\kappa }{\oplus }{y}_2\right)=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x,y_1\right)y_2^j{a}_j\left(\kappa \right).$$

We note that the following series manipulation formulas hold:

$$\sum _{M=0}^{\infty }g\left(M\right)\frac{{(x_1+x_2)}^M}{M!}=\sum _{l,j=0}^{\infty }g(l+j)\frac{x_1^l}{l!}\frac{x_2^j}{j!}$$

(26)

and

$$\sum _{n,m=0}^{\infty }B\left(m,n\right)=\sum _{n=0}^{\infty }\sum _{m=0}^{n}B\left(m,n-m\right).$$

(27)

The following theorem concerns a binary-finite summation formula for the $\kappa$-Bell polynomials.

Theorem 8.

(Implicit summation formula) we have

$${\varphi }_{j+l,\kappa }\left(x,y\right)=\sum _{n,m=0}^{j,l}\left(\genfrac{}{}{0pt}{}{j}{n}\right)\left(\genfrac{}{}{0pt}{}{l}{m}\right){\varphi }_{j+l-n-m,\kappa }\left(x,w\right)a_{n+m}\left(\kappa \right){\left(y\stackrel{\kappa }{\ominus }w\right)}^{n+m}.$$

(28)

Proof. 

Replacing t by $t+u$ in (20), we get

$$e^{x\left({exp}_{\kappa }\left(t+u\right)-1\right)}={exp}_{\kappa }\left(-y\left(t+u\right)\right)\sum _{j,l=0}^{\infty }{\varphi }_{j+l,\kappa }\left(x,y\right)\frac{t^j}{j!}\frac{u^l}{l!}.$$

Again, by changing w by y in the previous equality, and utilizing (26), it is obtained that

$$e^{x\left({exp}_{\kappa }\left(t+u\right)-1\right)}={exp}_{\kappa }\left(-w\left(t+u\right)\right)\sum _{j,l=0}^{\infty }{\varphi }_{j+l,\kappa }\left(x,w\right)\frac{t^j}{j!}\frac{u^l}{l!}.$$

Using (4), it is examined from two previous equalities that

$$\sum _{n=0}^{\infty }{\varphi }_{j+l,\kappa }\left(x,y\right)\frac{t^j}{j!}\frac{u^l}{l!}={exp}_{\kappa }\left(\left(y\stackrel{\kappa }{\ominus }w\right)\left(t+u\right)\right)\sum _{j,l=0}^{\infty }{\varphi }_{j+l,\kappa }\left(x,w\right)\frac{t^j}{j!}\frac{u^l}{l!},$$

which means

$$\sum _{n=0}^{\infty }{\varphi }_{j+l,\kappa }\left(x,y\right)\frac{t^j}{j!}\frac{u^l}{l!}=\sum _{n,m=0}^{\infty }{a}_{n+m}\left(\kappa \right){\left(y\stackrel{\kappa }{\ominus }w\right)}^{n+m}\frac{t^n}{n!}\frac{u^m}{m!}\sum _{j,l=0}^{\infty }{\varphi }_{j+l,\kappa }\left(x,w\right)\frac{t^j}{j!}\frac{u^l}{l!}.$$

Hence, it is derived from (27) that

$$\sum _{n=0}^{\infty }{\varphi }_{j+l,\kappa }\left(x,y\right)\frac{t^j}{j!}\frac{u^l}{l!}=\sum _{j,l=0}^{\infty }\sum _{n,m=0}^{j,l}\frac{a_{n+m}\left(\kappa \right){\left(y\stackrel{\kappa }{\ominus }w\right)}^{n+m}{\varphi }_{j+l-n-m,\kappa }\left(x,w\right)}{n!m!\left(j-n\right)!\left(l-m\right)!}{t}^j{u}^l,$$

which results in the alleged consequence (28). □

Corollary 5.

It can be analyzed from (28) that

$${\varphi }_{j,\kappa }\left(x,y\right)=\sum _{n=0}^j\left(\genfrac{}{}{0pt}{}{j}{n}\right){\varphi }_{j-n,\kappa }\left(x,w\right)a_n\left(\kappa \right){\left(y\stackrel{\kappa }{\ominus }w\right)}^n.$$

Corollary 6.

It can be analyzed from (28) that

$${\varphi }_{j,\kappa }\left(x,y\stackrel{\kappa }{\oplus }w\right)=\sum _{n=0}^j\left(\genfrac{}{}{0pt}{}{j}{n}\right){\varphi }_{j-n,\kappa }\left(x,w\right)a_n\left(\kappa \right)y^n.$$

The summation of the products of two bivariate $\kappa$-Bell polynomials can be expressed in the form of a symmetric identity.

Theorem 9.

(Symmetric identity) The following identity

$$\begin{array}{ccc}& & \sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x,by\right){\varphi }_{j,\kappa }\left(x,ay\right)a^{n-j}{b}^j\hfill \\ & =& \sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{j,\kappa }\left(x,by\right){\varphi }_{n-j,\kappa }\left(x,ay\right)a^j{b}^{n-j}\hfill \end{array}$$

(29)

holds for $0\le n$ and $a,b\in \mathbb{R}.$

Proof. 

Let

$$\mathrm{{\rm Y}}=e^{x\left({exp}_{\kappa }\left(at\right)-1\right)}{e}^{x\left({exp}_{\kappa }\left(bt\right)-1\right)}{exp}_{\kappa }^2\left(yabt\right),$$

which is symmetric in a and b, and we examine that

$$\begin{array}{ccc}\hfill \mathrm{{\rm Y}}& =& \left(\sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x,by\right)\frac{{\left(at\right)}^n}{n!}\right)\left(\sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x,ay\right)\frac{{\left(bt\right)}^n}{n!}\right)\hfill \\ & =& \sum _{n=0}^{\infty }\left(\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)a^{n-j}{b}^j{\varphi }_{n-j,\kappa }\left(x,by\right){\varphi }_{j,\kappa }\left(x,ay\right)\right)\frac{t^n}{n!}\hfill \end{array}$$

and similarly

$$\mathrm{{\rm Y}}=\sum _{n=0}^{\infty }\left(\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)a^j{b}^{n-j}{\varphi }_{j,\kappa }\left(x,by\right){\varphi }_{n-j,\kappa }\left(x,ay\right)\right)\frac{t^n}{n!},$$

which give the desired result (29). □

5. $\mathsf{\kappa }$-Bernoulli and $\mathsf{\kappa }$-Euler Polynomials

In this part, we define Kaniadakis extensions of Bernoulli and Euler polynomials and develop diverse relations covering addition formulas and implicit summation formulas.

The classical Bernoulli $B_n\left(x\right)$ and Euler $E_n\left(x\right)$ polynomials (cf. [18]) are defined by the following generating functions:

$$\sum _{n=0}^{\infty }{B}_n\left(x\right)\frac{t^n}{n!}=\frac{t}{e^t-1}{e}^{xt},\left(\left|t\right|<2\pi \right),\sum _{n=0}^{\infty }{E}_n\left(x\right)\frac{t^n}{n!}=\frac{2}{e^t+1}{e}^{xt},\left(\left|t\right|<\pi \right).$$

(30)

The corresponding numbers of $B_n\left(x\right)$ and $E_n\left(x\right)$ are obtained by $x=0$ in (30). We now introduce the $\kappa$-Bernoulli and $\kappa$-Euler polynomials, as follows.

Definition 3.

The κ-Bernoulli polynomials of order $\alpha \in \mathbb{R}$ and κ-Euler polynomials of order $\alpha \in \mathbb{R}$ are defined by means of the following generating functions about $t=0$, respectively, as follows:

$$\sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)\frac{t^n}{n!}={\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\alpha }{exp}_{\kappa }\left(xt\right),\left|t\right|<\left|{ln}_{\kappa }\left(e^{2\pi }\right)\right|$$

(31)

and

$$\sum _{n=0}^{\infty }{E}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)\frac{t^n}{n!}={\left(\frac{2}{{exp}_{\kappa }\left(t\right)+1}\right)}^{\alpha }{exp}_{\kappa }\left(xt\right),\left|t\right|<\left|{ln}_{\kappa }\left(e^{\pi }\right)\right|.$$

(32)

The corresponding numbers of $B_{n,\kappa }^{\left(\alpha \right)}\left(x\right)$ and $E_{n,\kappa }^{\left(\alpha \right)}\left(x\right)$ are obtained by choosing $x=0$ in (31) and (32), namely $B_{n,\kappa }^{\left(\alpha \right)}:=B_{n,\kappa }^{\left(\alpha \right)}\left(0\right)$ and $E_{n,\kappa }^{\left(\alpha \right)}:=E_{n,\kappa }^{\left(\alpha \right)}\left(0\right)$. Also, upon setting $\alpha =1$ in (31) and (32), the polynomials $B_{n,\kappa }^{\left(\alpha \right)}\left(x\right)$ and $E_{n,\kappa }^{\left(\alpha \right)}\left(x\right)$ become the $\kappa$-Bernoulli polynomials $B_{n,\kappa }\left(x\right)$ and the $\kappa$-Euler polynomials $E_{n,\kappa }\left(x\right)$ given by

$$\sum _{n=0}^{\infty }{B}_{n,\kappa }\left(x\right)\frac{t^n}{n!}=\frac{t{exp}_{\kappa }\left(xt\right)}{{exp}_{\kappa }\left(t\right)-1}\mathrm{and}\sum _{n=0}^{\infty }{E}_{n,\kappa }\left(x\right)\frac{t^n}{n!}=\frac{2{exp}_{\kappa }\left(xt\right)}{{exp}_{\kappa }\left(t\right)+1}.$$

(33)

The corresponding numbers of $B_{n,\kappa }\left(x\right)$ and $E_{n,\kappa }\left(x\right)$ are obtained by choosing $x=0$ in (33), namely $B_{n,\kappa }:=B_{n,\kappa }\left(0\right)$ and $E_{n,\kappa }:=E_{n,\kappa }\left(0\right)$.

Remark 7.

In the limit as $\kappa \to 0$, the polynomials $B_{n,\kappa }^{\left(\alpha \right)}\left(x\right)$ and $E_{n,\kappa }^{\left(\alpha \right)}\left(x\right)$ reduce to their classical counterparts:

$$\underset{\kappa \to 0}{lim}{B}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)=B_n^{\left(\alpha \right)}\left(x\right),\underset{\kappa \to 0}{lim}{E}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)=E_n^{\left(\alpha \right)}\left(x\right).$$

The $\kappa$-Bernoulli polynomials of order $\alpha$ can be represented by the $\kappa$-Bernoulli numbers of order $\alpha$, as stated in the following theorem.

Theorem 10.

The following formula for κ-Bernoulli polynomials of order α holds for $n\ge 0$ and $\alpha \in \mathbb{R}$:

$$B_{n,\kappa }^{\left(\alpha \right)}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }^{\left(\alpha \right)}{a}_l\left(\kappa \right)x^l.$$

(34)

Proof. 

We observe from (5) and (31) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)\frac{t^n}{n!}& =& {\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\alpha }{exp}_{\kappa }\left(xt\right)\hfill \\ & =& \sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha \right)}\frac{t^n}{n!}\sum _{n=0}^{\infty }{a}_n\left(\kappa \right)x^n\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }^{\left(\alpha \right)}{a}_l\left(\kappa \right)x^l\right)\frac{t^n}{n!}.\hfill \end{array}$$

Thus, we arrive at the claimed formula (34). □

The $\kappa$-Bernoulli polynomials of order $\alpha +\beta$ can be represented as sums of products of the $\kappa$-Bernoulli polynomials of orders $\alpha$ and $\beta$ by the following theorem.

Theorem 11.

The following formula for κ-Bernoulli polynomials of order α holds for $n\ge 0$ and $\alpha ,\beta \in \mathbb{R}$:

$$B_{n,\kappa }^{\left(\alpha +\beta \right)}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }^{\left(\alpha \right)}\left(x\right)B_{l,\kappa }^{\left(\beta \right)}.$$

(35)

Proof. 

It follows from (31) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha +\beta \right)}\left(x\right)\frac{t^n}{n!}& =& {\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\alpha +\beta }{exp}_{\kappa }\left(xt\right)\hfill \\ & =& {\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\alpha }{exp}_{\kappa }\left(xt\right){\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\beta }\hfill \\ & =& \sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)\frac{t^n}{n!}\sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\beta \right)}\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }^{\left(\alpha \right)}\left(x\right)B_{l,\kappa }^{\left(\beta \right)}\right)\frac{t^n}{n!}.\hfill \end{array}$$

Hence, we obtain the assertion (35). □

The $\kappa$-addition formula for $\kappa$-Bernoulli polynomials of order $\alpha$ can be written in terms of $\kappa$-Bernoulli polynomials of order $\alpha$ by the following theorem.

Theorem 12.

The following formula for κ-Bernoulli polynomials of order α holds for $n\ge 0$ and $\alpha \in \mathbb{R}$:

$$B_{n,\kappa }^{\left(\alpha \right)}\left(x\stackrel{\kappa }{\oplus }y\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }^{\left(\alpha \right)}\left(x\right)a_l\left(\kappa \right)y^l.$$

(36)

Proof. 

It can be observed from (4) and (31) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha \right)}\left(x\stackrel{\kappa }{\oplus }y\right)\frac{t^n}{n!}& =& {\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\alpha }{exp}_{\kappa }\left(\left(x\stackrel{\kappa }{\oplus }y\right)t\right)\hfill \\ & =& {\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\alpha }{exp}_{\kappa }\left(xt\right){exp}_{\kappa }\left(yt\right)\hfill \\ & =& \sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)\frac{t^n}{n!}\sum _{n=0}^{\infty }{a}_n\left(\kappa \right)y^n\frac{t^n}{n!}\hfill \\ & & \sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }^{\left(\alpha \right)}\left(x\right)a_l\left(\kappa \right)y^l\right)\frac{t^n}{n!}.\hfill \end{array}$$

Therefore, we get the desired formula (36). □

The $\kappa$-Bernoulli polynomials of order $\alpha$ can be written using higher-order $\kappa$-Bernoulli polynomials as follows.

Theorem 13.

The following formula for κ-Bernoulli polynomials of order α holds for $n\ge 0$ and $\alpha \in \mathbb{R}$:

$$B_{n,\kappa }^{\left(\alpha \right)}\left(x\right)=\frac{1}{n+1}\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n+1}{l}\right)B_{l,\kappa }^{(\alpha +1)}\left(x\right)a_{n+1-l}\left(\kappa \right).$$

(37)

Proof. 

We see from (31) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)\frac{t^n}{n!}& =& \frac{{exp}_{\kappa }\left(t\right)-1}{t}{\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\alpha +1}{exp}_{\kappa }\left(xt\right)\hfill \\ & =& \frac{{exp}_{\kappa }\left(t\right)}{t}{\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\alpha +1}{exp}_{\kappa }\left(xt\right)-\frac{1}{t}{\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{\alpha +1}{exp}_{\kappa }\left(xt\right)\hfill \\ & =& \sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha +1\right)}\left(x\right)\frac{t^n}{n!}\sum _{n=0}^{\infty }{a}_n\left(\kappa \right)\frac{t^{n-1}}{n!}-\sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha +1\right)}\left(x\right)\frac{t^{n-1}}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{l,\kappa }^{\left(\alpha +1\right)}\left(x\right)a_{n-l}\left(\kappa \right)\right)\frac{t^{n-1}}{n!}-\sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(\alpha +1\right)}\left(x\right)\frac{t^{n-1}}{n!}.\hfill \end{array}$$

Hence, we acquire the asserted formula (37). □

Corollary 7.

Upon setting $\alpha =1$ in (34), (36), and (37), we get the formulas for κ-Bernoulli polynomials as follows:

$$\begin{array}{c}{B}_{n,\kappa }\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }{a}_l\left(\kappa \right)x^l,\\ B_{n,\kappa }\left(x\stackrel{\kappa }{\oplus }y\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }\left(x\right)a_l\left(\kappa \right)y^l,\\ a_n\left(\kappa \right)x^n=\frac{1}{n+1}\left(\sum _{l=0}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{l}\right)B_{l,\kappa }\left(x\right)a_{n+1-l}\left(\kappa \right)-B_{n+1,\kappa }\left(x\right)\right).\end{array}$$

(38)

Remark 8.

We notice that the identity (38) is a generalization of the well-known identity for the familiar Bernoulli polynomials stated below (cf. [18]):

$$x^n=\frac{1}{n+1}\sum _{l=0}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{l}\right)B_k\left(x\right)-B_{n+1}\left(x\right).$$

The following theorems on the $\kappa$-Euler polynomials of order $\alpha$ are stated without proof, since their derivation parallels that of the $\kappa$-Bernoulli polynomials of order $\alpha$.

Theorem 14.

The following formula for κ-Euler polynomials of order α holds for $n\ge 0$ and $\alpha \in \mathbb{R}$:

$$E_{n,\kappa }^{\left(\alpha \right)}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)E_{n-l,\kappa }^{\left(\alpha \right)}{x}^l{a}_l\left(\kappa \right).$$

(39)

Proof. 

The proof of this theorem can be completed similarly to that of Theorem 10. So, we omit it. □

Theorem 15.

The following formula for κ-Euler polynomials of order α holds for $n\ge 0$ and $\alpha ,\beta \in \mathbb{R}$:

$$E_{n,\kappa }^{\left(\alpha +\beta \right)}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)E_{n-l,\kappa }^{\left(\alpha \right)}\left(x\right)E_{l,\kappa }^{\left(\beta \right)}.$$

Proof. 

The proof of this theorem can be completed similarly to that of Theorem 11. So, we omit it. □

Theorem 16.

The following formula for κ-Euler polynomials of order α holds for $n\ge 0$ and $\alpha \in \mathbb{R}$:

$$E_{n,\kappa }^{\left(\alpha \right)}\left(x\stackrel{\kappa }{\oplus }y\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)E_{n-l,\kappa }^{\left(\alpha \right)}\left(x\right)y^l{a}_l\left(\kappa \right).$$

(40)

Proof. 

The proof of this theorem can be completed similarly to that of Theorem 12. So, we omit it. □

Theorem 17.

The following formula for κ-Euler polynomials of order α holds for $n\ge 0$ and $\alpha \in \mathbb{R}$:

$$E_{n,\kappa }^{\left(\alpha \right)}\left(x\right)=\frac{1}{2}\left(\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)E_{l,\kappa }^{\left(\alpha +1\right)}\left(x\right)a_{n-l}\left(\kappa \right)+E_{n,\kappa }^{\left(\alpha +1\right)}\left(x\right)\right).$$

(41)

Proof. 

The proof of this theorem can be completed similarly to that of Theorem 13. So, we omit it. □

Corollary 8.

Upon setting $\alpha =1$ in (39), (40), and (41), we get the formulas for Kaniadakis Euler polynomials as follows.

$$\begin{array}{c}{E}_{n,\kappa }\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)E_{n-l,\kappa }{a}_l\left(\kappa \right)x^l,\\ E_{n,\kappa }\left(x\stackrel{\kappa }{\oplus }y\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)E_{n-l,\kappa }\left(x\right)a_l\left(\kappa \right)y^l,\\ a_n\left(\kappa \right)x^n=\frac{1}{2}\left(\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)E_{l,\kappa }\left(x\right)a_{n-l}\left(\kappa \right)+E_{n,\kappa }\left(x\right)\right).\end{array}$$

(42)

Remark 9.

We notice that the identity (42) is a generalization of the well-known identity for the usual Euler polynomials stated below (cf. [18]):

$$x^n=\frac{1}{2}\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)E_l\left(x\right)+E_n\left(x\right).$$

6. Some Connected Formulas

In this section, we develop several formulas and correlations covering the $\kappa$-Euler polynomials, the $\kappa$-Bernoulli polynomials, the $\kappa$-Stirling polynomials of the second kind, and the $\kappa$-Bell polynomials.

Below, we give a fundamental property including the $\kappa$-Bell polynomials and $\kappa$-Stirling polynomials of the second kind.

Theorem 18.

The following correlation including ${\varphi }_{n,\kappa }\left(x,y\right)$ and $S_{2,\kappa }\left(n,j:x\right)$

$${\varphi }_{n,\kappa }\left(x,y\right)=\sum _{j=0}^n{x}^j{S}_{2,\kappa }\left(n,j:y\right)$$

(43)

holds for $j,n\in {\mathbb{N}}_0$ with $j\le n$.

Proof. 

We observe from (10) and (20) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{\varphi }_{n,\kappa }\left(x,y\right)\frac{t^n}{n!}& =& e^{x\left({exp}_{\kappa }\left(t\right)-1\right)}{exp}_{\kappa }\left(yt\right)\hfill \\ & =& \sum _{j=0}^{\infty }{x}^j\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}{exp}_{\kappa }\left(yt\right)\hfill \\ & =& \sum _{j=0}^{\infty }{x}^j\sum _{n=0}^{\infty }{S}_{2,\kappa }\left(n,j:y\right)\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\left(\sum _{j=0}^n{S}_{2,\kappa }\left(n,j:y\right)x^j\right)\frac{t^n}{n!},\hfill \end{array}$$

which gives the alleged consequence (43). □

Corollary 9.

Some special cases of the relation (43) are as follows:

$${\varphi }_{n,\kappa }\left(x\right)=\sum _{j=0}^n{x}^j{S}_{2,\kappa }\left(n,j\right),$$

(44)

$${\varphi }_{n,\kappa }\left(1,y\right)=\sum _{j=0}^n{S}_{2,\kappa }\left(n,j:y\right),$$

and

$${\varphi }_{n,\kappa }=\sum _{j=0}^n{S}_{2,\kappa }\left(n,j\right).$$

Remark 10.

The relation (44) is a κ-analog of the well-known formula in (19).

A relation including the $\kappa$-Euler polynomials and $\kappa$-Stirling polynomials of the second kind is provided below.

Theorem 19.

The following correlation including $E_{n,\kappa }\left(x\right)$ and $S_{2,\kappa }\left(n,j:x\right)$

$$E_{n,\kappa }\left(x\right)=\sum _{j=0}^n{\left(-2\right)}^{-j}j!S_{2,\kappa }\left(n,j:x\right)$$

(45)

holds for $j\le n$.

Proof. 

We observe from (10) and (32) that

$$\begin{array}{ccc}\hfill \sum _{n=0}^{\infty }{E}_{n,\kappa }\left(x\right)\frac{t^n}{n!}& =& {exp}_{\kappa }\left(xt\right)\frac{2}{{exp}_{\kappa }\left(t\right)+1}={exp}_{\kappa }\left(xt\right){\left(\frac{{exp}_{\kappa }\left(t\right)-1}{2}+1\right)}^{-1}\hfill \\ & =& \sum _{j=0}^{\infty }{\left(-1\right)}^j{\left(\frac{{exp}_{\kappa }\left(t\right)-1}{2}\right)}^j{exp}_{\kappa }\left(xt\right)\hfill \\ & =& \sum _{j=0}^{\infty }{\left(-2\right)}^{-j}j!\sum _{n=0}^{\infty }{S}_{2,\kappa }\left(n,j:x\right)\frac{t^n}{n!}\hfill \\ & =& \sum _{n=0}^{\infty }\left(\sum _{j=0}^n{\left(-2\right)}^{-j}j!S_{2,\kappa }\left(n,j:x\right)\right)\frac{t^n}{n!},\hfill \end{array}$$

which means the alleged consequence (45). □

Corollary 10.

We have a relation including the κ-Euler numbers and κ-Stirling numbers of the second kind:

$$E_{n,\kappa }=\sum _{j=0}^n{\left(-2\right)}^{-j}j!S_{2,\kappa }\left(n,j\right).$$

Here, we provide some connected formulas by the consecutive theorems with their proofs.

Theorem 20.

The following summation formula

$$B_{n,\kappa }^{\left(-j\right)}\left(x\right)=\frac{n!j!}{\left(n+j\right)!}{S}_{2,\kappa }\left(n+j,j:x\right)$$

(46)

is valid for $j<n$.

Proof. 

In terms of (10) and (32), we see that

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{B}_{n,\kappa }^{(-j)}\left(x\right)\frac{t^n}{n!}& ={\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{-j}{exp}_{\kappa }\left(xt\right)\hfill \\ \hfill & =j!t^{-j}\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}{exp}_{\kappa }\left(xt\right)\hfill \\ \hfill & =j!\sum _{n=j}^{\infty }{S}_{2,\kappa }(n,j;x)\frac{t^{n-j}}{n!},\hfill \end{array}$$

which means the alleged consequence (46). □

Theorem 21.

The following formula

$$\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }^{\left(j\right)}\left(x\right)S_{2,\kappa }\left(l,j:y\right)=\left(\genfrac{}{}{0pt}{}{n}{j}\right){\left(x\stackrel{\kappa }{\oplus }y\right)}^{n-j}$$

(47)

holds for $n>j$.

Proof. 

In terms of (10) and (32), we see that

$$\begin{array}{cc}\hfill & \left(\sum _{n=0}^{\infty }{B}_{n,\kappa }^{\left(j\right)}\left(x\right)\frac{t^n}{n!}\right)\left(\sum _{n=j}^{\infty }{S}_{2,\kappa }(n,j;y)\frac{t^n}{n!}\right)\hfill \\ \hfill & ={\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^j{exp}_{\kappa }\left(xt\right)·\frac{{\left({exp}_{\kappa }\left(t\right)-1\right)}^j}{j!}{exp}_{\kappa }\left(yt\right)\hfill \\ \hfill & =\frac{t^j}{j!}{exp}_{\kappa }\left(\left(x\stackrel{\kappa }{\oplus }y\right)t\right)\hfill \\ \hfill & =\frac{1}{j!}\sum _{n=0}^{\infty }{\left(x\stackrel{\kappa }{\oplus }y\right)}^n\frac{t^{n+j}}{n!},\hfill \end{array}$$

which means the alleged consequence (47). □

Theorem 22.

The following formula

$${\varphi }_{n+l,\kappa }\left(x,y\right)=\frac{\left(n+l\right)!}{n!}\sum _{l=0}^{\infty }\frac{x^l}{l!}{B}_{n,\kappa }^{\left(-l\right)}\left(y\right)$$

(48)

is valid for $n\in {\mathbb{N}}_0$.

Proof. 

In terms of (20) and (32), we see that

$$\begin{array}{cc}\hfill \sum _{n=0}^{\infty }{\varphi }_{n,\kappa }(x,y)\frac{t^n}{{\kappa }_n!}& =e^{\left(x\left({exp}_{\kappa }\left(t\right)-1\right)\right)}{exp}_{\kappa }\left(yt\right)\hfill \\ \hfill & =\sum _{l=0}^{\infty }\frac{x^l}{l!}{\left({exp}_{\kappa }\left(t\right)-1\right)}^l{exp}_{\kappa }\left(yt\right)\hfill \\ \hfill & =\sum _{l=0}^{\infty }\frac{x^l}{l!}\sum _{n=0}^{\infty }{B}_{n,\kappa }^{(-l)}\left(y\right)\frac{t^{n+l}}{n!}\hfill \\ \hfill & =\sum _{n=0}^{\infty }\sum _{l=0}^{\infty }\frac{x^l}{l!}{B}_{n,\kappa }^{(-l)}\left(y\right)\frac{t^{n+l}}{n!},\hfill \end{array}$$

which means the alleged consequence (48). □

7. Further Remarks

Here, we provide some representations and formulas for the new polynomials defined in the previous sections.

7.1. p-Adic Integral Representations

We give p-adic integral representations for the polynomials defined in the previous sections. We first review the notations of p-adic integrals; see [19,20]. In this section, we need the following definitions and notations.

Let ${\mathbb{Z}}_p$ be a set of p-adic integers. Let $\mathbb{K}$ be a field with a complete valuation and $C^1({\mathbb{Z}}_p\to \mathbb{K})$ be a set of continuous derivative functions. That is, $C^1({\mathbb{Z}}_p\to \mathbb{K})$ is contained in the following set

$$\left\{f:X\to \mathbb{K}:f\left(x\right)\mathrm{is}\mathrm{differentiable}\mathrm{and}\frac{d}{dx}f\left(x\right)\mathrm{is}\mathrm{continuous}\right\}.$$

The Volkenborn integral (p-adic bosonic integral) of the function

$$f\in C^1({\mathbb{Z}}_p\to \mathbb{K})$$

is given by

$${\int }_{{\mathbb{Z}}_p}f\left(x\right)d\mu \left(x\right)=\underset{N\to \infty }{lim}\frac{1}{p^N}\sum _{x=0}^{p^N-1}f\left(x\right).$$

(49)

We note that $\mu \left(x\right)=\mu \left(x+p^N{\mathbb{Z}}_p\right)$ is the Haar distribution, defined by (cf. [19,20]; see also the references cited in each of these earlier works):

$$\mu \left(x+p^N{\mathbb{Z}}_p\right)=\frac{1}{p^N}.$$

Assume that $C\left({\mathbb{Z}}_p\right)$ is the space of all continuous functions on ${\mathbb{Z}}_p$. For $f\in C\left({\mathbb{Z}}_p\right)$, the fermionic p-adic integral on ${\mathbb{Z}}_p$ is given by Kim (see [19]), as follows:

$$I_{-1}\left(f\right)={\int }_{{\mathbb{Z}}_p}f\left(y\right)d{\mu }_{-1}\left(y\right)=\underset{N\to \infty }{lim}\frac{1}{p^N}\sum _{l=0}^{p^N-1}{\left(-1\right)}^{l}f\left(l\right).$$

(50)

Upon setting $f\left(t\right)=e^{\left(x+y\right)t}$ in (49) and (50), we acquire (see [19])

$$B_n\left(x\right)={\int }_{{\mathbb{Z}}_p}{\left(x+y\right)}^{n}d\mu \left(y\right)\mathrm{and}{E}_n\left(x\right)={\int }_{{\mathbb{Z}}_p}{\left(x+y\right)}^{n}d{\mu }_{-1}\left(y\right),$$

(51)

which are the familiar Bernoulli and Euler polynomials, respectively, in (30). Letting $x=0$ in (51), we obtain

$$B_n={\int }_{{\mathbb{Z}}_p}{y}^{n}d\mu \left(y\right)\mathrm{and}{E}_n={\int }_{{\mathbb{Z}}_p}{y}^{n}d{\mu }_{-1}\left(y\right),$$

(52)

which are the familiar Bernoulli and Euler numbers, respectively.

We give the following Corollaries.

Corollary 11.

Utilizing (12) and (52), the Volkenborn integral and the fermionic p-adic integral representations of the κ-Stirling polynomials of the second kind are provided by

$${\int }_{{\mathbb{Z}}_p}{S}_{2,\kappa }\left(n,j:x\right)d\mu \left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)S_{2,\kappa }\left(n-l,j\right)a_l\left(\kappa \right)B_l,$$

and

$${\int }_{{\mathbb{Z}}_p}{S}_{2,\kappa }\left(n,j:x\right)d{\mu }_{-1}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)S_{2,\kappa }\left(n-l,j\right)a_l\left(\kappa \right)E_l.$$

Corollary 12.

Using (24) and (52), the Volkenborn integral and the fermionic p-adic integral representations of the bivariate κ-Bell polynomials are provided by

$${\int }_{{\mathbb{Z}}_p}{\varphi }_{n,\kappa }\left(x,y\right)d\mu \left(y\right)=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x\right)a_j\left(\kappa \right)B_j,$$

and

$${\int }_{{\mathbb{Z}}_p}{\varphi }_{n,\kappa }\left(x,y\right)d{\mu }_{-1}\left(y\right)=\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right){\varphi }_{n-j,\kappa }\left(x\right)a_j\left(\kappa \right)E_j.$$

Corollary 13.

By means of (34) and (52), the Volkenborn integral and the fermionic p-adic integral representations of the κ-Bernoulli polynomials are provided by

$${\int }_{{\mathbb{Z}}_p}{B}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)d\mu \left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }^{\left(\alpha \right)}{a}_l\left(\kappa \right)B_l,$$

and

$${\int }_{{\mathbb{Z}}_p}{B}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)d{\mu }_{-1}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{n-l,\kappa }^{\left(\alpha \right)}{a}_l\left(\kappa \right)E_l.$$

Corollary 14.

Utilizing (38) and (52), we obtain the following relations, including old and new polynomials

$$B_n=\frac{{\sum }_{l=0}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{l}\right)\left({\sum }_{j=0}^l\left(\genfrac{}{}{0pt}{}{l}{j}\right)B_{l-j,\kappa }{a}_j\left(\kappa \right)B_j{a}_{n+1-l}\left(\kappa \right)-B_{n+1-l,\kappa }{a}_l\left(\kappa \right)B_l\right)}{a_n\left(\kappa \right)\left(n+1\right)},$$

and

$$E_n=\frac{{\sum }_{l=0}^{n+1}\left(\genfrac{}{}{0pt}{}{n+1}{l}\right)\left({\sum }_{j=0}^l\left(\genfrac{}{}{0pt}{}{l}{j}\right)B_{l-j,\kappa }{a}_j\left(\kappa \right)E_j{a}_{n+1-l}\left(\kappa \right)-B_{n+1-l,\kappa }{a}_l\left(\kappa \right)E_l\right)}{a_n\left(\kappa \right)\left(n+1\right)}.$$

Corollary 15.

Using (39) and (52), the Volkenborn integral and the fermionic p-adic integral representations of the κ-Bernoulli polynomials are provided by

$${\int }_{{\mathbb{Z}}_p}{E}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)d\mu \left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)a_l\left(\kappa \right)E_{n-l,\kappa }^{\left(\alpha \right)}{B}_l,$$

and

$${\int }_{{\mathbb{Z}}_p}{E}_{n,\kappa }^{\left(\alpha \right)}\left(x\right)d{\mu }_{-1}\left(x\right)=\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)a_l\left(\kappa \right)E_{n-l,\kappa }^{\left(\alpha \right)}{E}_l.$$

Corollary 16.

Utilizing (42) and (52), we obtain the following relations, which involve both classical and new polynomials:

$$B_n=\frac{{\displaystyle \sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)a_{n-l}\left(\kappa \right)\sum _{j=0}^l\left(\genfrac{}{}{0pt}{}{l}{j}\right)a_j\left(\kappa \right)E_{l-j,\kappa }{B}_j+\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)a_j\left(\kappa \right)E_{n-j,\kappa }{B}_j}}{2a_n\left(\kappa \right)},$$

and

$$E_n=\frac{{\displaystyle \sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)a_{n-l}\left(\kappa \right)\sum _{j=0}^l\left(\genfrac{}{}{0pt}{}{l}{j}\right)a_j\left(\kappa \right)E_{l-j,\kappa }{E}_j+\sum _{j=0}^n\left(\genfrac{}{}{0pt}{}{n}{j}\right)a_j\left(\kappa \right)E_{n-j,\kappa }{E}_j}}{2a_n\left(\kappa \right)},$$

Example 1

(Case $n=2$). From the Corollary 16, we have

$$2a_2\left(\kappa \right)B_2=\sum _{l=0}^2\left(\genfrac{}{}{0pt}{}{2}{l}\right)a_{2-l}\left(\kappa \right)\sum _{j=0}^l\left(\genfrac{}{}{0pt}{}{l}{j}\right)a_j\left(\kappa \right)E_{l-j,\kappa }{B}_j+\sum _{j=0}^2\left(\genfrac{}{}{0pt}{}{2}{j}\right)a_j\left(\kappa \right)E_{2-j,\kappa }{B}_j.$$

First sum:

$$\begin{array}{cc}\hfill S_1& :=\sum _{l=0}^2\left(\genfrac{}{}{0pt}{}{2}{l}\right)a_{2-l}\left(\kappa \right)\sum _{j=0}^l\left(\genfrac{}{}{0pt}{}{l}{j}\right)a_j\left(\kappa \right)E_{l-j,\kappa }{B}_j\hfill \\ & =\underset{l=0}{\underbrace{\left(\genfrac{}{}{0pt}{}{2}{0}\right)a_2\left(\kappa \right)\left(\genfrac{}{}{0pt}{}{0}{0}\right)a_0\left(\kappa \right)E_{0,\kappa }{B}_0}}+\underset{l=1}{\underbrace{\left(\genfrac{}{}{0pt}{}{2}{1}\right)a_1\left(\kappa \right)\left(\left(\genfrac{}{}{0pt}{}{1}{0}\right)a_0\left(\kappa \right)E_{1,\kappa }{B}_0+\left(\genfrac{}{}{0pt}{}{1}{1}\right)a_1\left(\kappa \right)E_{0,\kappa }{B}_1\right)}}\hfill \\ & +\underset{l=2}{\underbrace{\left(\genfrac{}{}{0pt}{}{2}{2}\right)a_0\left(\kappa \right)\left(\left(\genfrac{}{}{0pt}{}{2}{0}\right)a_0\left(\kappa \right)E_{2,\kappa }{B}_0+\left(\genfrac{}{}{0pt}{}{2}{1}\right)a_1\left(\kappa \right)E_{1,\kappa }{B}_1+\left(\genfrac{}{}{0pt}{}{2}{2}\right)a_2\left(\kappa \right)E_{0,\kappa }{B}_2\right)}}\hfill \\ & =1+2\left((-\frac{1}{2})+(-\frac{1}{2})\right)+\left(0+2·\frac{1}{4}+B_2\right)=B_2-\frac{1}{2}.\hfill \end{array}$$

Second sum:

$$\begin{array}{cc}\hfill S_2& :=\sum _{j=0}^2\left(\genfrac{}{}{0pt}{}{2}{j}\right)a_j{E}_{2-j,\kappa }{B}_j=\left(\genfrac{}{}{0pt}{}{2}{0}\right)a_0{E}_{2,\kappa }{B}_0+\left(\genfrac{}{}{0pt}{}{2}{1}\right)a_1{E}_{1,\kappa }{B}_1+\left(\genfrac{}{}{0pt}{}{2}{2}\right)a_2{E}_{0,\kappa }{B}_2\hfill \\ & =0+2·(-\frac{1}{2})·(-\frac{1}{2})+B_2=B_2+\frac{1}{2}.\hfill \end{array}$$

Therefore,

$$2a_2\left(\kappa \right)B_2=S_1+S_2=(B_2-\frac{1}{2})+(B_2+\frac{1}{2})=2B_2,$$

and since $a_2\left(\kappa \right)=1$, the identity is satisfied. By applying the classical Bernoulli relation ${\sum }_{j=0}^n\left(\genfrac{}{}{0pt}{}{n+1}{j}\right)B_j=0$ for $n=2$, we additionally obtain

$$B_2=\frac{1}{6}.$$

Similarly, we easily obtain $E_2$.

Corollary 17.

In view of (24) and (52), some other p-adic integral representations of the bivariate κ-Bell polynomials are provided by

$${\int }_{{\mathbb{Z}}_p}{\varphi }_{n,\kappa }\left(x,y\right)d\mu \left(x\right)=\sum _{j=0}^n{B}_j{S}_{2,\kappa }\left(n,j:y\right),$$

and

$${\int }_{{\mathbb{Z}}_p}{\varphi }_{n,\kappa }\left(x,y\right)d{\mu }_{-1}\left(x\right)=\sum _{j=0}^n{E}_j{S}_{2,\kappa }\left(n,j:y\right).$$

7.2. Determinantal Representations

Determinantal forms of special polynomials provide deep structural insight and computational utility. For instance, Qi and Guo derived Hessenberg determinant expressions for Bernoulli polynomials in [21]. More recently, Cao et al. ([22]) introduced determinantal identities for the differences between Bernoulli polynomials and numbers. Wani and Nahid dealt with the derivation of determinant and integral forms for the two iterated 2D Appell polynomials in [23]. These representations not only facilitate the explicit computation of higher-order terms by means of determinant evaluation, but also unify a wide variety of polynomial families under a common linear algebraic perspective.

We now provide determinantal representations for the $\kappa$-Bernoulli polynomials and the $\kappa$-Euler polynomials.

Let

$$\mathrm{\Omega }\left(t\right)=\sum _{n=0}^{\infty }{\delta }_n\frac{t^n}{n!}={\left(\frac{t}{{exp}_{\kappa }\left(t\right)-1}\right)}^{-1},$$

(53)

where ${\delta }_n$ is a sequence. Then, we derive from (33) and (53) that

$$\left(\sum _{n=0}^{\infty }{B}_{n,\kappa }\frac{t^n}{n!}\right)\mathrm{\Omega }\left(t\right)=\sum _{n=0}^{\infty }\left(\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{l,\kappa }{\delta }_{n-l}\right)\frac{t^n}{n!},$$

which yields

$$\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{l,\kappa }{\delta }_{n-l}=\left\{\begin{array}{c}1\mathrm{for}n=0,\\ 0\mathrm{for}n>0.\end{array}\right.$$

Therefore, we get

$$\left\{\begin{array}{c}{\delta }_0=\frac{1}{B_{0,\kappa }},\hfill \\ {\displaystyle {\delta }_n=-\sum _{l=1}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{l,\kappa }{\delta }_{n-l},}\hfill & n\ge 1.\hfill \end{array}\right.$$

We give a determinantal representation for the $\kappa$-Bernoulli polynomials as follows.

Theorem 23.

We have

$$B_{0,\kappa }\left(x\right)=\frac{1}{{\delta }_0}$$

and

$$B_{n,\kappa }\left(x\right)=\frac{{\left(-1\right)}^n}{{\left({\delta }_0\right)}^{n+1}}\left|\begin{array}{cccccc}1& x& a_2\left(\kappa \right)x^2& \cdots & a_{n-1}\left(\kappa \right)x^{n-1}& a_n\left(\kappa \right)x^n\\ {\delta }_0& {\delta }_1& {\delta }_2& \cdots & {\delta }_{n-1}& {\delta }_n\\ 0& {\delta }_0& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\delta }_1& \cdots & \left(\genfrac{}{}{0pt}{}{n-1}{1}\right){\delta }_{n-2}& \left(\genfrac{}{}{0pt}{}{n}{1}\right){\delta }_{n-1}\\ 0& 0& \ddots & & \left(\genfrac{}{}{0pt}{}{n-1}{2}\right){\delta }_{n-3}& \left(\genfrac{}{}{0pt}{}{n}{2}\right){\delta }_{n-2}\\ ⋮& ⋮& & \ddots & ⋮& ⋮\\ ⋮& ⋮& & & & \\ 0& 0& \cdots & \cdots & {\delta }_0& \left(\genfrac{}{}{0pt}{}{n}{n-1}\right){\delta }_1\end{array}\right|.$$

(54)

Proof. 

It can be observed from (33) and (53) that

$$\sum _{n=0}^{\infty }{a}_n\left(\kappa \right)x^n\frac{t^n}{n!}=\sum _{n=0}^{\infty }\sum _{l=0}^n\left(\genfrac{}{}{0pt}{}{n}{l}\right)B_{l,\kappa }\left(x\right){\delta }_{n-l}\frac{t^n}{n!},$$

which gives the following infinite system of equations in the unknown variables:

$$\begin{array}{ccc}\hfill a_0\left(\kappa \right)x^0& =& B_{0,\kappa }\left(x\right){\delta }_0,\hfill \\ \hfill a_1\left(\kappa \right)x^1& =& B_{0,\kappa }\left(x\right){\delta }_1+B_{1,\kappa }\left(x\right){\delta }_0,\hfill \\ & & ⋮\hfill \\ \hfill a_n\left(\kappa \right)x^n& =& B_{0,\kappa }\left(x\right){\delta }_n+\left(\genfrac{}{}{0pt}{}{n}{1}\right)B_{1,\kappa }\left(x\right){\delta }_{n-1}+\cdots +B_{n,\kappa }\left(x\right){\delta }_0.\hfill \end{array}$$

Because of the specific structure of the aforementioned system (lower triangular), we can determine the unknown variables $B_{n,\kappa }\left(x\right)$ by exclusively using the first $n+1$ equations. This can be achieved by employing Cramer’s rule, which facilitates the computation of the solution:

$$B_{n,\kappa }\left(x\right)=\frac{1}{{\left({\delta }_0\right)}^{n+1}}\left|\begin{array}{cccccc}{\delta }_0& 0& 0& 0& \cdots & a_0\left(\kappa \right)\\ {\delta }_1& {\delta }_0& 0& 0& \cdots & a_1\left(\kappa \right)x^1\\ {\delta }_2& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\delta }_1& {\delta }_0& 0& \cdots & a_2\left(\kappa \right)x^2\\ ⋮& ⋮& & \ddots & & ⋮\\ {\delta }_{n-1}& \left(\genfrac{}{}{0pt}{}{n-1}{1}\right){\delta }_{n-2}& \left(\genfrac{}{}{0pt}{}{n-2}{2}\right){\delta }_{n-3}& \cdots & \cdots & a_{n-1}\left(\kappa \right)x^{n-1}\\ {\delta }_n& \left(\genfrac{}{}{0pt}{}{n}{1}\right){\delta }_{n-1}& \left(\genfrac{}{}{0pt}{}{n}{2}\right){\delta }_{n-2}& \left(\genfrac{}{}{0pt}{}{n}{3}\right){\delta }_{n-3}& \cdots & a_n\left(\kappa \right)x^n\end{array}\right|,$$

which also can be rewritten as

$$B_{n,\kappa }\left(x\right)=\frac{1}{{\left({\delta }_0\right)}^{n+1}}\left|\begin{array}{cccccc}{\delta }_0& {\delta }_1& {\delta }_2& \cdots & {\delta }_{n-1}& {\delta }_n\\ 0& {\delta }_0& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\delta }_1& \cdots & \left(\genfrac{}{}{0pt}{}{n-1}{1}\right){\delta }_{n-2}& \left(\genfrac{}{}{0pt}{}{n}{1}\right){\delta }_{n-1}\\ 0& 0& \ddots & & \left(\genfrac{}{}{0pt}{}{n-1}{2}\right){\delta }_{n-3}& \left(\genfrac{}{}{0pt}{}{n}{2}\right){\delta }_{n-2}\\ ⋮& ⋮& & \ddots & ⋮& ⋮\\ ⋮& ⋮& & & & \\ 0& 0& \cdots & \cdots & {\delta }_0& \left(\genfrac{}{}{0pt}{}{n}{n-1}\right){\delta }_1\\ 1& x& a_2\left(\kappa \right)x^2& \cdots & a_{n-1}\left(\kappa \right)x^{n-1}& a_n\left(\kappa \right)x^n\end{array}\right|.$$

Hence, we complete the proof. □

Example 2.

By Theorem 23, we have

$$B_{0,\kappa }\left(x\right)=1,$$

$$B_{1,\kappa }\left(x\right)=-\left|\begin{array}{cc}1& x\\ 1& \frac{1}{2}\end{array}\right|,$$

$$B_{2,\kappa }\left(x\right)=\left|\begin{array}{ccc}1& x& x^2\\ 1& \frac{1}{2}& \frac{1-{\kappa }^2}{6}\\ 0& 1& 1\end{array}\right|,$$

$$B_{3,\kappa }\left(x\right)=-\left|\begin{array}{cccc}1& x& x^2& \left(1-{\kappa }^2\right)x^3\\ 1& \frac{1}{2}& \frac{1-{\kappa }^2}{6}& \frac{1-4{\kappa }^2}{24}\\ 0& 1& 1& \frac{1-{\kappa }^2}{2}\\ 0& 0& 1& \frac{3}{2}\end{array}\right|,$$

$$B_{4,\kappa }\left(x\right)=\left|\begin{array}{ccccc}1& x& x^2& \left(1-{\kappa }^2\right)x^3& \left(1-4{\kappa }^2\right)x^3\\ 1& \frac{1}{2}& \frac{1-{\kappa }^2}{6}& \frac{1-4{\kappa }^2}{24}& \frac{\left(1-{\kappa }^2\right)\left(1-4{\kappa }^2\right)}{120}\\ 0& 1& 1& \frac{1-{\kappa }^2}{2}& \frac{1-4{\kappa }^2}{6}\\ 0& 0& 1& 3& 1-{\kappa }^2\\ 0& 0& 0& 1& 2\end{array}\right|.$$

Using the determinant representation (54) and Example 2, it is not hard to list the $\kappa$-Bernoulli polynomial sequences.

Similarly, we provide the following theorem without its proof.

Theorem 24.

The κ-Euler polynomials possess the following determinantal representation:

$$E_{0,\kappa }\left(x\right)=\frac{1}{{\delta }_0}$$

and

$$E_{n,\kappa }\left(x\right)=\frac{{\left(-1\right)}^n}{{\left({\delta }_0\right)}^{n+1}}\left|\begin{array}{cccccc}1& x& a_2\left(\kappa \right)x^2& \cdots & a_{n-1}\left(\kappa \right)x^{n-1}& a_n\left(\kappa \right)x^n\\ {\delta }_0& {\delta }_1& {\delta }_2& \cdots & {\delta }_{n-1}& {\delta }_n\\ 0& {\delta }_0& \left(\genfrac{}{}{0pt}{}{2}{1}\right){\delta }_1& \cdots & \left(\genfrac{}{}{0pt}{}{n-1}{1}\right){\delta }_{n-2}& \left(\genfrac{}{}{0pt}{}{n}{1}\right){\delta }_{n-1}\\ 0& 0& \ddots & & \left(\genfrac{}{}{0pt}{}{n-1}{2}\right){\delta }_{n-3}& \left(\genfrac{}{}{0pt}{}{n}{2}\right){\delta }_{n-2}\\ ⋮& ⋮& & \ddots & ⋮& ⋮\\ ⋮& ⋮& & & & \\ 0& 0& \cdots & \cdots & {\delta }_0& \left(\genfrac{}{}{0pt}{}{n}{n-1}\right){\delta }_1\end{array}\right|,$$

(55)

where the numerical sequence ${\left\{{\delta }_n\right\}}_{n\ge 0}$ is generated by

$$\sum _{n=0}^{\infty }{\delta }_n\frac{t^n}{n!}={\left(\frac{2}{{exp}_{\kappa }\left(t\right)+1}\right)}^{-1}.$$

Example 3.

By Theorem 23, we have

$$E_{0,\kappa }\left(x\right)=1,$$

$$E_{1,\kappa }\left(x\right)=-\left|\begin{array}{cc}1& x\\ 1& \frac{1}{2}\end{array}\right|,$$

$$E_{2,\kappa }\left(x\right)=\left|\begin{array}{ccc}1& x& x^2\\ 1& \frac{1}{2}& \frac{1}{4}\\ 0& 1& 1\end{array}\right|,$$

$$E_{3,\kappa }\left(x\right)=-\left|\begin{array}{cccc}1& x& x^2& \left(1-{\kappa }^2\right)x^3\\ 1& \frac{1}{2}& \frac{1}{4}& \frac{1-{\kappa }^2}{12}\\ 0& 1& 1& \frac{3}{4}\\ 0& 0& 1& 3\end{array}\right|,$$

$$E_{4,\kappa }\left(x\right)=\left|\begin{array}{ccccc}1& x& x^2& \left(1-{\kappa }^2\right)x^3& \left(1-4{\kappa }^2\right)x^3\\ 1& \frac{1}{2}& \frac{1}{4}& \frac{1-{\kappa }^2}{12}& \frac{1-4{\kappa }^2}{48}\\ 0& 1& 1& \frac{3}{4}& \frac{1-{\kappa }^2}{3}\\ 0& 0& 1& \frac{3}{2}& \frac{3}{2}\\ 0& 0& 0& 1& 2\end{array}\right|.$$

By the determinantal representation (55) and Example 3, it is not hard to list the $\kappa$-Euler polynomial sequences.

8. Conclusions and Observation

Recently, Kaniadakis calculus (cf. [1]) has attracted significant attention. Its main features arise from the $\kappa$-deformed exponential function

$${exp}_{\kappa }\left(t\right)={\left(\sqrt{1+{\kappa }^2{t}^2}+\kappa t\right)}^{1/\kappa },0\le \kappa <1,$$

which provides a continuous one-parameter deformation of the classical exponential function. After the introduction of the $\kappa$-algebra, the associated $\kappa$-differential and $\kappa$-integral calculus has been established, covering not only the $\kappa$-exponential and $\kappa$-logarithm but also $\kappa$-versions of other fundamental functions of ordinary mathematics (cf. [4,5,7]).

In this paper, we have introduced and investigated several new classes of $\kappa$-deformed special polynomials. We have also defined the $\kappa$-Stirling polynomials of the second kind and derived explicit relations, including closed-form expressions, summation formulas, and addition formulas. We have considered the generating function of the bivariate $\kappa$-Bell polynomials and established various properties such as summation formulas, addition formulas, symmetric identities, and implicit summations. Furthermore, we have introduced the $\kappa$-Bernoulli polynomials of order $\alpha$ and the $\kappa$-Euler polynomials of order $\alpha$, deriving a range of identities that connect these families with the $\kappa$-Stirling and $\kappa$-Bell polynomials. In addition, we have provided p-adic Volkenborn and p-adic fermionic integral representations for these families and discussed determinantal representations for the $\kappa$-Bernoulli and $\kappa$-Euler polynomials, which lead to the construction of further new families and reinforce the structural results obtained.

In conclusion, the $\kappa$-extensions of Bernoulli, Euler, Stirling, and Bell-type families represent natural generalizations associated with the $\kappa$-exponential function. By embedding these families within the framework of $\kappa$-calculus, we have established novel structural insights that parallel the behavior of $\kappa$-distributions and $\kappa$-transforms. It would be of further interest to investigate additional families of special functions, numbers, and polynomials constructed within the $\kappa$-calculus perspective. We believe that this direction of research will not only enrich the theory of special functions but also stimulate new applications in mathematical physics, engineering, probability, and statistics.