Soft rough fuzzy sets based on covering

R. Mareay; Radwan Abu-Gdairi; M. Badr; R. Mareay; Radwan Abu-Gdairi; M. Badr

doi:10.3934/math.2024548

AIMS Mathematics

2024, Volume 9, Issue 5: 11180-11193. doi: 10.3934/math.2024548

Previous Article Next Article

Research article Special Issues

Soft rough fuzzy sets based on covering

R. Mareay ^{1
,
,},
Radwan Abu-Gdairi ^{2
,},
M. Badr ^{3
,}

1.
Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafrelsheikh 33516, Egypt; Email: roshdeymareay@sci.kfs.edu.eg
2.
Department of Mathematics, Faculty of Science, Zarqa University, Zarqa 13132, Jordan; Email: rgdairi@zu.edu.jo
3.
Department of Mathematics, Faculty of Science, New Valley University, Egypt; Email: m.shaban@sci.nvu.edu.eg

Received: 30 December 2023 Revised: 08 March 2024 Accepted: 15 March 2024 Published: 21 March 2024
MSC : 54A40, 03E72, 54C08

Soft rough fuzzy sets ( $\mathcal SRFSs$ ) represent a powerful paradigm that integrates soft computing, rough set theory, and fuzzy logic. This research aimed to comprehensively investigate the various dimensions of $\mathcal SRFSs$ within the domain of approximation structures. The study encompassed a wide spectrum of concepts, ranging from covering approximation structures and soft rough coverings to soft neighborhoods, fuzzy covering approximation operators, and soft fuzzy covering approximation operators. We introduced three models of $\mathcal SRFSs$ based on covering via the core of soft neighborhood. We discussed and analyzed our models' characteristics and properties. The relations between our models for soft fuzzy covering sets and Zhan's model for soft rough fuzzy covering were presented.

Keywords:

Citation: R. Mareay, Radwan Abu-Gdairi, M. Badr. Soft rough fuzzy sets based on covering[J]. AIMS Mathematics, 2024, 9(5): 11180-11193. doi: 10.3934/math.2024548

Related Papers:

[1]	Sarker Ashraful Islam, Farhana Kabir Esheta, Md Mahir Shahriar, Dewan Hasan Ahmed . Numerical study of aerodynamic drag reduction of a circular cylinder with an inbuilt nozzle. Metascience in Aerospace, 2024, 1(4): 379-400. doi: 10.3934/mina.2024018
[2]	Hussein Kokash, G. Gilou Agbaglah . A two- to three-dimensional wake transition mechanism induced by the angle of attack of a NACA0012 airfoil. Metascience in Aerospace, 2024, 1(3): 329-345. doi: 10.3934/mina.2024015
[3]	Jubilee Prasad Rao, Sai Naveen Chimata . Machine learning-based surrogates for eVTOL performance prediction and design optimization. Metascience in Aerospace, 2024, 1(3): 246-267. doi: 10.3934/mina.2024011
[4]	Hui Ying WANG, Némo DECAMPS . Effects of oxygen concentration of oxidizer flow on laminar diffusion flame established over liquid fuel beds at microgravity. Metascience in Aerospace, 2024, 1(2): 159-184. doi: 10.3934/mina.2024007
[5]	Pasynok Sergey . Cumulative STF coefficients evaluation and validation. Metascience in Aerospace, 2024, 1(4): 371-378. doi: 10.3934/mina.2024017
[6]	Dulce M Graciano, Fernando Z Sierra-Espinosa, Juan C García . Numerical simulation of vortex-induced vibration in a bladeless turbine: Effects of separation distance between tandem harvesters. Metascience in Aerospace, 2024, 1(3): 309-328. doi: 10.3934/mina.2024014
[7]	James H. Page, Lorenzo Dorbolò, Marco Pretto, Alessandro Zanon, Pietro Giannattasio, Michele De Gennaro . Sensitivity analysis of mixed analysis-synthesis flight profile reconstruction. Metascience in Aerospace, 2024, 1(4): 401-415. doi: 10.3934/mina.2024019
[8]	Xu Han, Bin Yu, Hong Liu . Mixing mechanisms in the view of mixing indicators: from passive-scalar mixing to variable-density mixing. Metascience in Aerospace, 2024, 1(1): 1-37. doi: 10.3934/mina.2024001
[9]	Weizhen Tang, Jie Dai, Zhousheng Huang . Comprehensive assessment of air pollutant and noise emissions at airports across different altitudes. Metascience in Aerospace, 2024, 1(3): 292-308. doi: 10.3934/mina.2024013
[10]	Yuan Hu, Zhihao Jiang, Xintai Yuan, Xifan Hua, Wei Liu . Isometric mapping algorithm based GNSS-R sea ice detection. Metascience in Aerospace, 2024, 1(1): 38-52. doi: 10.3934/mina.2024002

Abstract

1. Introduction and preliminaries

We begin with the following definitions of notations:

$\begin{equation*} \mathbb{N = }\left\{ 1, 2, 3, \cdots \right\} \text{ and }\mathbb{N}_{0}: = \mathbb{ N\cup }\left\{ 0\right\} \text{.} \end{equation*}$

Also, as usual, $\mathbb{R}$ denotes the set of real numbers and $\mathbb{C}$ denotes the set of complex numbers.

The two variable Laguerre polynomials $L_{n}(u, v)$ ^[1] are defined by the Taylor expansion about $\tau = 0$ (also popularly known as generating function) as follows:

$\begin{equation*} \sum\limits_{p = 0}^{\infty }L_{p}(u, v)\frac{\tau ^{p}}{p!} = e^{v\tau }C_{0}(u\tau ), \end{equation*}$

where is the $0$ -th order Tricomi function ^[19] given by

$\begin{equation*} C_{0}(u) = \sum\limits_{p = 0}^{\infty }\frac{(-1)^{p}u^{p}}{(p!)^{2}} \end{equation*}$

and has the series representation

$\begin{equation*} L_{p}(u, v) = \sum\limits_{s = 0}^{p}\; \frac{p!(-1)^{s}v^{p-s}u^{s}}{(p-s)!(s!)^{2}}. \end{equation*}$

The classical Euler polynomials $E_{p}(u)$ , Genocchi polynomials $G_{p}(u)$ and the Bernoulli polynomials $B_{p}(u)$ are usually defined by the generating functions (see, for details and further work, ^{[1,2,4,5,6,7,9,11,12,20]}):

$\begin{equation*} \sum\limits_{p = 0}^{\infty }E_{p}(u)\frac{\tau ^{p}}{p!} = \frac{2}{e^{\tau }+1} e^{u\tau }\quad (|\tau | \lt \pi ), \end{equation*}$

$\begin{equation*} \sum\limits_{p = 0}^{\infty }G_{p}(u)\frac{\tau ^{p}}{p!} = \frac{2\tau }{e^{\tau }+1} e^{u\tau }\quad (|\tau | \lt \pi ) \end{equation*}$

and

$\begin{equation*} \sum\limits_{p = 0}^{\infty }B_{p}(u)\frac{\tau ^{p}}{p!} = \frac{\tau }{e^{\tau }-1} e^{u\tau }\quad (|\tau | \lt 2\pi ). \end{equation*}$

The Daehee polynomials, recently originally defined by Kim et al. ^[9], are defined as follows

$\begin{equation} \sum\limits_{p = 0}^{\infty }D_{p}(u)\frac{\tau ^{p}}{p!} = \frac{\log (1+\tau )}{\tau } (1+\tau )^{u}, \end{equation}$

(1.1)

where, for $u = 0$ , $D_{p}(0) = D_{p}$ stands for Daehee numbers given by

$\begin{equation} \sum\limits_{p = 0}^{\infty }D_{p}\, \frac{\tau ^{p}}{p!} = \frac{\log (1+\tau )}{\tau }. \end{equation}$

(1.2)

Due to Kim et al.'s idea ^[9], Jang et al. ^[3] gave the partially degenarate Genocchi polynomials as follows:

$\begin{equation} \frac{2\log (1+\tau \lambda )^{\frac{1}{\lambda }}}{e^{\tau }+1}e^{u\tau } = \sum\limits_{p = 0}^{\infty }G_{p, \lambda }(u)\frac{\tau ^{p}}{p!}, \end{equation}$

(1.3)

which, for the case $u = 0$ , yields the partially degenerate Genocchi numbers $G_{n, \lambda }: = G_{n, \lambda }(0)$ .

Pathan et al. ^[17] considered the generalization of Hermite-Bernoulli polynomials of two variables ${}_{H}B_{p}^{(\alpha)}(u, v)$ as follows

$\begin{equation} \left( \frac{\tau }{e^{\tau }-1}\right) ^{\alpha }e^{u\tau +v\tau ^{2}} = \sum\limits_{p = 0}^{\infty }{}_{H}B_{p}^{(\alpha )}(u, v)\frac{\tau ^{p}}{p!}. \end{equation}$

(1.4)

On taking $\alpha = 1$ in (1.4) yields a well known result of [2,p. 386 (1.6)] given by

$\begin{equation} \left( \frac{\tau }{e^{\tau }-1}\right) e^{u\tau +v\tau ^{2}} = \sum\limits_{p = 0}^{\infty }{}_{H}B_{p}(u, v)\frac{\tau ^{p}}{p!}. \end{equation}$

(1.5)

The two variable Laguerre-Euler polynomials (see ^[7,8]) are defined as

$\begin{equation} \sum\limits_{p = 0}^{\infty }{}_{L}E_{p}(u, v)\frac{\tau ^{p}}{p!} = \frac{2}{e^{\tau }+1}e^{v\tau }C_{0}(u\tau ). \end{equation}$

(1.6)

The alternating sum $T_{k}(p)$ , where $k\in \mathbb{N}_{0}$ , (see ^[14]) is given as

$\begin{equation*} T_{k}(p) = \sum\limits_{j = 0}^{p}(-1)^{j}j^{k} \end{equation*}$

and possess the generating function

$\begin{equation} \sum\limits_{r = 0}^{\infty }T_{k}(p)\frac{\tau ^{r}}{r!} = \frac{1-(-e^{\tau })^{(p+1)} }{e^{\tau }+1}. \end{equation}$

(1.7)

The idea of degenerate numbers and polynomials found existence with the study related to Bernoulli and Euler numbers and polynomials. Lately, many researchers have begun to study the degenerate versions of the classical and special polynomials (see ^{[3,10,11,12,13,14,15,16,18]}, for a systematic work). Influenced by their works, we introduce partially degenerate Laguerre-Genocchi polynomials and also a new generalization of partially degenerate Laguerre-Genocchi polynomials and then give some of their applications. We also derive some implicit summation formula and general symmetry identities.

2. Partially degenerate Laguerre-Genocchi polynomials

Let $\lambda, \tau \in \mathbb{C}$ with $|\tau \lambda |\leq 1$ and $\tau \lambda \neq -1$ . We introduce and investigate the partially degenerate Laguerre-Genocchi polynomials as follows:

$\begin{equation} \sum\limits_{p = 0}^{\infty }{}_{L}G_{p, \lambda }(u, v)\frac{\tau ^{p}}{p!} = \frac{ 2\log (1+\lambda \tau )^{\frac{1}{\lambda }}}{e^{\tau }+1}e^{v\tau }C_{0}(u\tau ). \end{equation}$

(2.1)

In particular, when $\lambda\to 0$ , ${}_LG_{p, \lambda}(u, v)\to {}_LG_p(u, v)$ and they have the closed form given as

$\begin{equation*} {}_{L}G_{p, \lambda }(u, v) = \sum\limits_{q = 0}^{p}\binom{p}{q}G_{q, \lambda }L_{p-q}(u, v). \end{equation*}$

Clearly, $u = 0$ in (2.1) gives ${}_{L}G_{p, \lambda }(0, 0): = G_{p, \lambda }$ that stands for the partially degenerate Genocchi polynomials ^[3].

Theorem 1. For $p \in \mathbb{N}_{o}$ , the undermentioned relation holds:

$\begin{equation} {}_LG_{p, \lambda}(u, v) = \sum\limits_{q = 0}^{p} \binom{p}{q+1}\, q!(-\lambda)^{q}{}_LG_{p-q-1}(u, v). \end{equation}$

(2.2)

Proof. With the help of (2.1), one can write

$\begin{align*} \sum\limits_{p = 0}^{\infty }{}_{L}G_{p, \lambda }(u, v)\frac{\tau ^{p}}{p!} = & \frac{ 2\log (1+\lambda \tau )^{\frac{1}{\lambda }}}{e^{\tau }+1}e^{v\tau }C_{0}(u\tau ) \\ = & \tau\left\{ \sum\limits_{q = 0}^{\infty }\frac{(-1)^{q}}{q+1}(\lambda \tau )^{q}\right\} \left\{ \sum\limits_{p = 0}^{\infty }{}_{L}G_{p}(u, v)\frac{\tau ^{p}}{p! }\right\} \\ = & \sum\limits_{p = 0}^{\infty }\left\{ \sum\limits_{q = 0}^{p}\binom{p}{q}\frac{(-\lambda )^{q}}{q+1}q!{}_{L}G_{p-q}(u, v)\right\} \frac{\tau ^{p+1}}{p!}, \end{align*}$

where, ${}_{L}G_{p-q}(u, v)$ are the Laguerre-Genocchi polynomials (see ^[8]). Finally, the assertion easily follows by equating the coefficients $\frac{ \tau ^{p}}{p!}$ .

Theorem 2. For $p\in \mathbb{N}_{o}$ , the undermentioned relation holds:

$\begin{equation} {}_{L}G_{p+1, \lambda }(u, v) = \sum\limits_{q = 0}^{p}\binom{p}{q}\lambda ^{q}(p+1){}_{L}G_{p-q+1}(u, v)D_{q}. \end{equation}$

(2.3)

Proof. We first consider

$\begin{align*} I_{1} = \frac{1}{\tau }\frac{2\log (1+\lambda \tau )^{\frac{1}{\lambda }}}{ e^{\tau }+1}e^{v\tau }C_{0}(u\tau ) = & \left\{ \sum\limits_{q = 0}^{\infty }D_{q}\frac{ (\lambda \tau )^{q}}{q!}\right\} \left\{ \sum\limits_{p = 0}^{\infty }{}_{L}G_{p}(u, v) \frac{\tau ^{p}}{p!}\right\} \\ = & \sum\limits_{p = 1}^{\infty }\left\{ \sum\limits_{q = 0}^{p}\binom{p}{q}(\lambda )^{q} D_{q}{}_{L}G_{p-q}(u, v)\right\} \frac{\tau ^{p}}{p!}. \end{align*}$

Next we have,

$\begin{align*} I_{2} = \frac{1}{\tau}\frac{2\log(1+\lambda \tau)^{\frac{1}{\lambda}}}{ e^{\tau}+1}e^{v\tau}C_{o}(u\tau) = &\frac{1}{\tau}\sum\limits_{p = 0}^{\infty}{}_LG_{p, \lambda}(u, v)\frac{\tau^{p}}{p!} \\ = &\sum\limits_{p = 0}^{\infty}\frac{{}_LG_{p+1, \lambda}(u, v)}{p+1}\frac{\tau^{p}}{p!}. \end{align*}$

Since $I_{1} = I_{2}$ , we conclude the assertion (2.3) of Theorem 2.

Theorem 3. For $p\in \mathbb{N}_{0}$ , the undermentioned relation holds:

$\begin{equation} {}_{L}G_{p, \lambda }(u, v) = p\sum\limits_{q = 0}^{p-1}\binom{p-1}{q}(\lambda )^{q}{}_{L}E_{p-q-1}(u, v)D_{q}. \end{equation}$

(2.4)

Proof. With the help of (2.1), one can write

$\begin{align*} \sum\limits_{p = 0}^{\infty }{}_{L}G_{p, \lambda }(x, y)\frac{\tau ^{p}}{p!} = & \left\{ \frac{\tau \log (1+\lambda \tau )}{\lambda \tau }\right\} \left\{ \frac{2}{ e^{\tau }+1}e^{v\tau }C_{0}(u\tau )\right\} \\ = & \tau \left\{ \sum\limits_{q = 0}^{\infty }D_{q}\frac{(\lambda \tau )^{q}}{q!} \right\} \left\{ \sum\limits_{p = 0}^{\infty }{}_{L}E_{p}(u, v)\frac{\tau ^{p}}{p!} \right\} \\ = & \sum\limits_{p = 0}^{\infty }\left\{ \sum\limits_{q = 0}^{p}\binom{p}{q}(\lambda )^{q}D_{q}\, {}_{L}E_{p-q}(u, v)\right\} \frac{\tau ^{p+1}}{p!}. \end{align*}$

Finally, the assertion (2.4) straightforwardly follows by equating the coefficients of same powers of $\tau$ above.

Theorem 4. For $p\in \mathbb{N}_{o}$ , the following relation holds:

$\begin{equation} {}_{L}G_{p, \lambda }(u, v+1) = \sum\limits_{q = 0}^{p}\binom{p}{q}{}\; _{L}G_{p-q, \lambda }(u, v). \end{equation}$

(2.5)

Proof. Using (2.1), we find

$\begin{align*} \sum\limits_{p = 0}^{\infty }& \left\{ {}_{L}G_{p, \lambda }(u, v+1)-{}_{L}G_{p, \lambda }(u, v)\right\}\frac{\tau^p}{p!} = \frac{2\log (1+\lambda \tau )^{\frac{1}{\lambda }}}{e^{\tau }+1} \\ \times & e^{(v+1)\tau }C_{0}(u\tau )-\frac{2\log (1+\lambda \tau )^{\frac{1}{ \lambda }}}{e^{\tau }+1}e^{v\tau }C_{0}(u\tau ) \\ = & \sum\limits_{p = 0}^{\infty }{}_{L}G_{p, \lambda }(u, v)\frac{\tau ^{p}}{p!} \sum\limits_{q = 0}^{\infty }\frac{\tau ^{q}}{q!}-\sum\limits_{p = 0}^{\infty }{}_{L}G_{p, \lambda }(u, v)\frac{\tau ^{p}}{p!} \\ = & \sum\limits_{p = 0}^{\infty }\left\{ \sum\limits_{q = 0}^{p}\binom{p}{q}\; {}_{L}G_{p-q, \lambda }(u, v)-{}_{L}G_{p, \lambda }(u, v)\right\} \frac{\tau ^{p}}{p!}. \end{align*}$

Hence, the assertion (2.5) straightforwardly follows by equating the coefficients of $\tau ^{p}$ above.

Theorem 5. For $p\in \mathbb{N}_{o}$ , the undermentioned relation holds:

$\begin{equation} {}_{L}G_{p, \lambda }(u, v) = \sum\limits_{q = 0}^{p}\sum\limits_{l = 0}^{q}\binom{p}{q}\binom{q}{l }G_{p-q}D_{q-l}\; \lambda ^{q-l}L_{l}(u, v). \end{equation}$

(2.6)

Proof. Since

$\begin{align*} \sum\limits_{p = 0}^{\infty }{}_{L}G_{p, \lambda }(u, v)\frac{\tau ^{p}}{p!} = & \frac{ 2\log (1+\lambda \tau )^{\frac{1}{\lambda }}}{e^{\tau }+1}e^{v\tau }C_{0}(u\tau ) \\ = & \left\{ \frac{2\tau }{e^{\tau }+1}\right\} \left\{ \frac{2\log (1+\lambda \tau )}{\lambda \tau }\right\} e^{v\tau }C_{0}(u\tau ) \\ = & \left\{ \sum\limits_{p = 0}^{\infty }G_{p}\frac{\tau ^{p}}{p!}\right\} \left\{ \sum\limits_{q = 0}^{\infty }D_{q}\frac{(\lambda \tau )^{q}}{q!}\right\} \left\{ \sum\limits_{l = 0}^{\infty }L_{l}(u, v)\frac{\tau ^{l}}{l!}\right\} , \end{align*}$

we have

$\begin{equation*} \sum\limits_{p = 0}^{\infty }{}_{L}G_{p, \lambda }(u, v)\frac{\tau ^{p}}{p!} = \sum\limits_{p = 0}^{\infty }\left\{ \sum\limits_{q = 0}^{p}\sum\limits_{l = 0}^{q}\binom{p}{q}\binom{q }{l}G_{p-q}D_{q-l}\lambda ^{q-l}L_{l}(u, v)\right\} \frac{\tau ^{p}}{p!}. \end{equation*}$

We thus complete the proof of Theorem 5.

Theorem 6. (Multiplication formula). For $p\in \mathbb{N}_{o}$ , the undermentioned relation holds:

$\begin{equation} {}_{L}G_{p, \lambda }(u, v) = f^{p-1}\sum\limits_{a = 0}^{f-1}{}_{L}G_{p, \frac{\lambda }{f }}\left( u, \frac{v+a}{f}\right). \; \; \; \; \; \; \; \; \end{equation}$

(2.7)

Proof. With the help of (2.1), we obtain

$\begin{align*} \sum\limits_{p = 0}^{\infty}{}_LG_{p, \lambda}(u, v)\frac{\tau^{p}}{p!} = &\frac{ 2\log(1+\lambda \tau)^{\frac{1}{\lambda}}}{e^{\tau}+1}e^{v\tau}C_{0}(u\tau) \\ = &\frac{2\log(1+\lambda \tau)^{\frac{1}{\lambda}}}{e^{\tau}+1} C_{0}(u\tau)\sum\limits_{a = 0}^{f-1}e^{(a+v)\tau} \\ = &\sum\limits_{p = 0}^{\infty}\left\lbrace f^{p-1}\sum\limits_{a = 0}^{f-1}{}_LG_{p, \frac{ \lambda}{f}}\left(u, \frac{v+a}{f}\right)\right\rbrace\frac{\tau^{p}}{p!}. \end{align*}$

Thus, the result in (2.7) straightforwardly follows by comparing the coefficients of $\tau^{p}$ above.

3. Generalized partially degenerate Laguerre-Genocchi polynomials

Consider a Dirichlet character $\chi$ and let $d\; \; (d\in \mathbb{N})$ be the conductor connected with it such that $d\equiv 1(\mod 2)$ (see ^[22]). Now we present a generalization of partially degenerate Laguerre-Genocchi polynomials attached to $\chi$ as follows:

$\begin{equation} \sum\limits_{p = 0}^{\infty }{}_{L}G_{p, \chi , \lambda }(u, v)\frac{\tau ^{p}}{p!} = \frac{2\log (1+\lambda \tau )^{\frac{1}{\lambda }}}{e^{f\tau }+1} \sum\limits_{a = 0}^{f-1}(-1)^{a}\chi (a)e^{(v+a)\tau }C_{0}(u\tau ). \end{equation}$

(3.1)

Here, $G_{p, \chi, \lambda } = {}_{L}G_{p, \chi, \lambda }(0, 0)$ are in fact, the generalized partially degenerate Genocchi numbers attached to the Drichlet character $\chi$ . We also notice that

$\underset{\begin{smallmatrix} \lambda \to 0 \\ v = 0 \end{smallmatrix}}{\mathop \lim\limits }\, \ {{\ }_{L}}{{G}_{p, \chi , \lambda }}(u, v) = {{G}_{p, \chi }}(u),$

is the familiar looking generalized Genocchi polynomial (see ^[20]).

Theorem 7. For $p\in \mathbb{N}_{0}$ , the following relation holds:

$\begin{equation} {}_{L}G_{p, \chi , \lambda }(u, v) = \sum\limits_{q = 0}^{p}\binom{p}{q}\lambda ^{q}D_{q}\, {}_{L}G_{p-q, \chi }(u, v). \end{equation}$

(3.2)

Proof. In view of (3.1), we can write

$\begin{equation*} \sum\limits_{p = 0}^{\infty}{}_LG_{p, \chi, \lambda}(u, v)\frac{\tau^{p}}{p!} = \frac{ 2\log(1+\lambda \tau)^{\frac{1}{\lambda}}}{e^{f\tau}+1} \sum\limits_{a = 0}^{f-1} (-1)^{a} \chi(a) e^{(v+a)\tau} C_{0}(u\tau) \end{equation*}$

$\begin{equation*} = \left\lbrace\frac{\log(1+\lambda \tau)}{\lambda \tau}\right\rbrace\left \lbrace \frac{2\tau}{e^{f\tau}+1}\sum\limits_{a = 0}^{f-1} (-1)^{a} \chi(a) e^{(v+a)\tau} C_{0}(u\tau)\right\rbrace \end{equation*}$

$\begin{equation*} = \left\lbrace\sum\limits_{q = 0}^{\infty}D_{q}\frac{\lambda^{q}}{\tau^{q}}{q!} \right\rbrace\left\lbrace \sum\limits_{p = 0}^{\infty}{}_LG_{p, \chi}(u, v)\frac{ \tau^{p}}{p!}\right\rbrace. \end{equation*}$

Finally, the assertion (3.2) of Theorem 7 can be achieved by equating the coefficients of same powers of $\tau$ .

Theorem 8. The undermentioned formula holds true:

$\begin{equation} {}_LG_{p, \chi, \lambda}(u, v) = f^{p-1} \sum\limits_{a = 0}^{f-1} (-1)^{a} \chi(a) {}_LG_{p, \frac{\lambda}{f}}\left(u, \frac{a+v}{f}\right). \end{equation}$

(3.3)

Proof. We first evaluate

$\begin{align*} \sum\limits_{p = 0}^{\infty}{}_LG_{p, \chi, \lambda}(u, v)\frac{\tau^{p}}{p!} = &\frac{ 2\log(1+\lambda \tau)^{\frac{1}{\lambda}}}{e^{f\tau}+1} \sum\limits_{a = 0}^{f-1} (-1)^{a} \chi(a) e^{(v+a)\tau} C_{0}(u\tau) \\ = &\frac{1}{f} \sum\limits_{a = 0}^{f-1} (-1)^{a} \chi(a) \frac{2\log(1+\lambda \tau)^{ \frac{f}{\lambda}}}{e^{f\tau}+1} e^{\left(\frac{a+v}{f}\right)f\tau} C_{0}(u\tau) \\ = &\sum\limits_{p = 0}^{\infty}\left\lbrace f^{p-1} \sum\limits_{a = 0}^{f-1} (-1)^{a} \chi(a) {}_LG_{p, \frac{\lambda}{f}}\left(u, \frac{a+v}{f}\right)\right\rbrace\frac{ \tau^{p}}{p!}. \end{align*}$

Now, the Theorem 8 can easily be concluded by equating the coefficients $\frac{\tau^{p}}{p!}$ above.

Using the result in (3.1) and with a similar approach used just as in above theorems, we provide some more theorems given below. The proofs are being omitted.

Theorem 9. The undermentioned formula holds true:

$\begin{equation} {}_{L}G_{p, \chi , \lambda }(u, v) = \sum\limits_{q = 0}^{p}\frac{G_{p-q, \chi , \lambda }(v)\; (-u)^{q}\; p!}{(q!)^{2}\, (p-q)!}. \end{equation}$

(3.4)

Theorem 10. The undermentioned formula holds true:

$\begin{equation} {}_{L}G_{p, \chi , \lambda }(u, v) = \sum\limits_{q = 0}^{p, l}\frac{G_{p-q-l, \chi , \lambda }\; (v)^{q}\; (-u)^{l}\; p!}{(p-q-l)!\; (q)!\; (l!)^{2}}. \end{equation}$

(3.5)

4. Implicit summation formulae

Theorem 11. The undermentioned formula holds true:

$\begin{equation} {}_{L}G_{l+h, \lambda }(u, \nu ) = \sum\limits_{p, n = 0}^{l, h}\binom{l}{p}\binom{h}{n} (u -v)^{p+n}{}_{L}G_{l+h-n-p, \lambda }(u, v). \end{equation}$

(4.1)

Proof. On changing $\tau$ by $\tau+\mu$ and rewriting (2.1), we evaluate

$\begin{equation*} e^{-v(\tau+\mu)} \sum\limits_{l, h = 0}^{\infty} {}_LG_{l+h, \lambda}(u, v) \frac{ \tau^{l}\mu^{h}}{l!h!} = \frac{2\log(1+\lambda (\tau+\mu))^{\frac{1}{\lambda}} }{e^{\tau+\mu}+1} C_{o}(u(\tau+\mu)), \end{equation*}$

which, upon replacing $v$ by $u$ and solving further, gives

$\begin{equation*} e^{(u-v)(\tau+\mu)} \sum\limits_{l, h = 0}^{\infty} {}_LG_{l+h, \lambda}(u, v) \frac{ \tau^{l}\mu^{h}}{l!h!} = \sum\limits_{l, h = 0}^{\infty} {}_LG_{l+h, \lambda}(u, \nu) \frac{\tau^{l}\mu^{h}}{l!h!}, \end{equation*}$

and also

$\begin{equation} \sum\limits_{P = 0}^{\infty} \frac{(u-v)^{P}(\tau+u)^{P}}{P!}\sum\limits_{l, h = 0}^{\infty} {}_LG_{l+h, \lambda}(u, v) \frac{\tau^{l}\mu^{h}}{l!h!} = \sum\limits_{l, h = 0}^{\infty} {}_LG_{l+h, \lambda}(u, \nu) \frac{\tau^{l}\mu^{h}}{l!h!}. \end{equation}$

(4.2)

Now applying the formula [21,p.52(2)]

$\begin{equation*} \sum\limits_{P = 0}^{\infty} f(P) \frac{(u+v)^{P}}{P!} = \sum\limits_{p, q = 0}^{\infty} f(p+q) \frac{u^{p}}{p!}\frac{v^{q}}{q!}, \end{equation*}$

in conjunction with (4.2), it becomes

$\begin{equation} \sum\limits_{p, n = 0}^{\infty} \frac{(u-v)^{p+n}\tau^{p}\mu^{n}}{p!n!} \sum\limits_{l, h = 0}^{\infty} {}_LG_{l+h, \lambda}(u, v) \frac{\tau^{l}\mu^{h}}{l!h!} = \sum\limits_{l, h = 0}^{\infty} {}_LG_{l+h, \lambda}(u, \nu) \frac{\tau^{l}\mu^{h}}{l!h! }.\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \end{equation}$

(4.3)

Further, upon replacing $l$ by $l-p$ , $h$ by $h-n$ , and using the result in [21,p.100 (1)], in the left of (4.3), we obtain

$\begin{equation*} \sum\limits_{p, n = 0}^{\infty} \sum\limits_{l, h = 0}^{\infty} \frac{(u-v)^{p+n}}{p!\, n!} {} _LG_{l+h-p-n, \lambda}(u, v) \frac{\tau^{l}\mu^{h}}{(l-p)!(h-n)!} = \sum\limits_{l, h = 0}^{\infty} {}_LG_{l+h, \lambda}(u, \nu) \frac{\tau^{l}\mu^{h}}{ l!\, h!}. \end{equation*}$

Finally, the required result can be concluded by equating the coefficients of the identical powers of $\tau^{l}$ and $\mu^{h}$ above.

Corollary 4.1. For $h = 0$ in (4.1), we get

$\begin{equation*} {}_{L}G_{l, \lambda }(u, \nu ) = \sum\limits_{\rho = 0}^{l}\binom{l}{\rho } \; {(u-v)^p}_{L}G_{l-\rho , \lambda }(u, v). \end{equation*}$

Some identities of Genocchi polynomials for special values of the parameters $u$ and $\nu$ in Theorem 11 can also be obtained. Now, using the result in (2.1) and with a similar approach, we provide some more theorems given below. The proofs are being omitted.

Theorem 12. The undermentioned formula holds good:

$\begin{equation*} {}_{L}G_{p, \lambda }(u, v+\mu ) = \sum\limits_{q = 0}^{p}\binom{p}{q}\mu ^{q}{}_{L}G_{p-q, \lambda }(u, v) \end{equation*}$

Theorem 13. The undermentioned implicit holds true:

$\begin{equation*} \sum\limits_{p = 0}^{\infty }{}_{L}G_{p, \lambda }(u, v)\frac{\tau ^{p}}{p!} = \frac{ 2\log (1+\lambda \tau )^{\frac{1}{\lambda }}}{e^{\tau }+1}e^{v\tau }C_{o}(u\tau ) = \sum\limits_{q = 0}^{p}\binom{p}{q}G_{p-q, \lambda }\; L_{p}(u, v) \end{equation*}$

and

$\begin{equation*} {}_{L}G_{p, \lambda }(u, v) = \sum\limits_{q = 0}^{p}\binom{p}{q}G_{p-q, \lambda }(u, v)\; L_{p}(u, v)\mathit{\text{.}} \end{equation*}$

Theorem 14. The undermentioned implicit summation formula holds:

$\begin{equation*} {}_{L}G_{p, \lambda }(u, v+1)+{}_{L}G_{p, \lambda }(u, v) = 2p\; \sum\limits_{q = 0}^{p-1} \binom{p-1}{q}\frac{(-\lambda )^{q}q!}{q+1}L_{p-q-1}(u, v). \end{equation*}$

Theorem 15. The undermentioned formula holds true:

$\begin{equation*} {}_LG_{p, \lambda}(u, v+1) = \sum\limits_{q = 0}^{p} {}_LG_{p-q, \lambda}(u, v). \end{equation*}$

5. Symmetry identities

Symmetry identities involving various polynomials have been discussed (e.g., ^{[7,9,10,11,17]}). As in above-cited work, here, in view of the generating functions (1.3) and (2.1), we obtain symmetry identities for the partially degenerate Laguerre-Genocchi polynomials ${}_{L}G_{n, \lambda }(u, v)$ .

Theorem 16. Let $\alpha, \beta \in \mathbb{Z}$ and $p\in \mathbb{N}_{0}$ , we have

$\begin{equation*} \sum\limits_{q = 0}^{p}\binom{p}{q}\beta ^{q}\alpha ^{p-q}{}_{L}G_{p-q, \lambda }(u\beta , v\beta ){}_{L}G_{q, \lambda }(u\alpha , v\alpha ) \end{equation*}$

$\begin{equation*} = \sum\limits_{q = 0}^{p}\binom{p}{q}\alpha ^{q}\beta ^{p-q}{}_{L}G_{p-q, \lambda }(u\alpha , v\alpha ){}_{L}G_{q, \lambda }(u\beta , v\beta )\mathit{\text{.}} \end{equation*}$

Proof. We first consider

$\begin{equation*} g(\tau ) = \frac{\left\{ 2\log (1+\lambda )^{\frac{\beta }{\lambda }}\right\} }{(e^{\alpha \tau }+1)}\frac{\left\{ 2\log (1+\lambda )^{\frac{\alpha }{ \lambda }}\right\} }{\; (e^{\beta \tau }+1)}e^{(\alpha +\beta )v\tau }C_{0}(u\alpha \tau )C_{0}(u\beta \tau ). \end{equation*}$

Now we can have two series expansion of $g(\tau)$ in the following ways:

On one hand, we have

$\begin{eqnarray} g(\tau ) & = &\left( \sum\limits_{p = 0}^{\infty }\; _{L}G_{p, \lambda }(u\beta , v\beta )\frac{\left( \alpha \tau \right) ^{p}}{p!}\right) \left( \sum\limits_{q = 0}^{\infty }\; _{L}G_{q, \lambda }(u\alpha , v\alpha )\frac{ \left( \beta \tau \right) ^{q}}{q!}\right) \\ & = &\sum\limits_{p = 0}^{\infty }\left( \sum\limits_{q = 0}^{p}\binom{p}{q} \beta ^{q}\alpha ^{p-q}\; _{L}G_{p-q, \lambda }(u\beta , v\beta )\; _{L}G_{q, \lambda }(u\alpha , v\alpha )\right) \frac{\tau ^{p}}{p!}. \end{eqnarray}$

(5.1)

and on the other, we can write

$\begin{eqnarray} g(\tau ) & = &\left( \sum\limits_{p = 0}^{\infty }\; _{L}G_{p, \lambda }(u\alpha , v\alpha )\frac{\left( \beta \tau \right) ^{p}}{p!}\right) \left( \sum\limits_{q = 0}^{\infty }\; _{L}G_{q, \lambda }(u\beta , v\beta )\frac{\left( \alpha \tau \right) ^{q}}{q!}\right) \\ & = &\sum\limits_{p = 0}^{\infty }\left( \sum\limits_{q = 0}^{p}\binom{p}{q} \alpha ^{q}\beta ^{p-q}\; _{L}G_{p-q, \lambda }(u\alpha , v\alpha )\; _{L}G_{q, \lambda }(u\beta , v\beta )\right) \frac{\tau ^{p}}{p!}. \end{eqnarray}$

(5.2)

Finally, the result easily follows by equating the coefficients of $\tau ^{p}$ on the right-hand side of Eqs (5.1) and (5.2).

Theorem 17. Let $\alpha, \beta \in \mathbb{Z}$ with $p\in \mathbb{N}_{0}$ , Then,

$\begin{equation*} \sum\limits_{q = 0}^{p}\binom{p}{q}\beta ^{q}\alpha ^{p-q}\sum\limits_{\sigma = 0}^{\alpha -1}\sum\limits_{\rho = 0}^{\beta -1}(-1)^{\sigma +\rho }{}_{L}G_{p-q, \lambda }(u, v\beta +\frac{\beta }{\alpha }\sigma +\rho )G_{q, \lambda }(z\alpha ) \end{equation*}$

$\begin{equation*} = \sum\limits_{q = 0}^{p}\binom{p}{q}\alpha ^{p}\beta ^{p-q}\sum\limits_{\sigma = 0}^{\beta -1}\sum\limits_{\rho = 0}^{\alpha -1}(-1)^{\sigma +\rho }{}_{L}G_{p-q, \lambda }(u, v\alpha +\frac{\beta }{\alpha }\sigma +\rho )G_{q, \lambda }(z\beta ) \mathit{\text{.}} \end{equation*}$

Proof. Let

$\begin{equation*} g(\tau) = \frac{\left\lbrace 2\log(1+\lambda)^{\frac{\alpha}{\lambda} }\right\rbrace}{(e^{\alpha\tau}+1)^{2}}\frac{\left\lbrace 2\log(1+\lambda)^{ \frac{\beta}{\lambda}}\right\rbrace}{(e^{\beta\tau}+1)^{2}} e^{(\alpha\beta\tau+1)^{2}} e^{(\alpha\beta)(v+z)\tau}[Cs_{0}(u\tau)]. \end{equation*}$

Considering $g(\tau)$ in two forms. Firstly,

$\begin{align*} g(\tau)& = \frac{\left\lbrace2\log(1+\lambda)^{\frac{\alpha}{\lambda} }\right\rbrace}{e^{\alpha\tau}+1}e^{\alpha\beta v\tau}C_{o}(u\tau) \left( \frac{e^{\alpha\beta\tau}+1}{e^{\beta\tau}+1}\right) \notag \\ &\times\frac{\left\lbrace2\log(1+\lambda)^{\frac{\beta}{\lambda} }\right\rbrace}{e^{\beta\tau}+1} e^{\alpha\beta z\tau} \left(\frac{ e^{\alpha\beta\tau}+1}{e^{\alpha\tau}+1}\right) \end{align*}$

$\begin{align} & = \frac{\left\{ 2\log (1+\lambda )^{\frac{\alpha }{\lambda }}\right\} }{ e^{\alpha \tau }+1}e^{\alpha \beta v\tau }C_{0}(u\tau )\left( \sum\limits_{\sigma = 0}^{\alpha -1}(-1)^{\sigma }e^{\beta \tau \sigma }\right) \\ & \times \frac{\left\{ 2\log (1+\lambda )^{\frac{\beta }{\lambda }}\right\} }{e^{\beta \tau }+1}e^{\alpha \beta \tau z}C_{0}(u\tau )\left( \sum\limits_{\rho = 0}^{\beta -1}(-1)^{\rho }e^{\alpha \tau \rho }\right) , \end{align}$

(5.3)

Secondly,

$\begin{align} & g(\tau ) \\ & = \sum\limits_{p = 0}^{\infty }\left\{ \sum\limits_{q = 0}^{p}\binom{p}{q}\; \beta ^{q}\alpha ^{p-q}\sum\limits_{\sigma = 0}^{\alpha -1}\sum\limits_{\rho = 0}^{\beta -1}(-1)^{\sigma +\rho }{}_{L}G_{p-q, \lambda }\left( u\alpha , v\beta +\frac{\beta }{\alpha } \sigma +\rho \right) G_{q, \lambda }(\alpha z)\right\} \frac{\tau ^{p}}{p!} \\ & = \sum\limits_{p = 0}^{\infty }\left\{ \sum\limits_{q = 0}^{p}\binom{p}{q}\; \alpha ^{q}\beta ^{p-q}\sum\limits_{\sigma = 0}^{\alpha -1}\sum\limits_{\rho = 0}^{\beta -1}(-1)^{\sigma +\rho }{}_{L}G_{\sigma -\rho , \lambda }\left( u, v\alpha +\frac{\alpha }{ \beta }\sigma +\rho \right) G_{q, \lambda }(z\beta )\right\} \frac{\tau ^{p}}{ p!}.\; \; \; \; \end{align}$

(5.4)

Finally, the result straightforwardly follows by equating the coefficients of $\tau^{p}$ in Eqs (5.3) and (5.4).

We now give the following two Theorems. We omit their proofs since they follow the same technique as in the Theorems 16 and 17.

Theorem 18. Let $\alpha, \beta \in \mathbb{Z}$ and $p\in \mathbb{N}_{0}$ , Then,

$\begin{eqnarray*} &&\sum\limits_{q = 0}^{p}\binom{p}{q}\beta ^{q}\alpha ^{p-q}\sum\limits_{\sigma = 0}^{\alpha -1}\sum\limits_{\rho = 0}^{\beta -1}(-1)^{\sigma +\rho }{}_{L}G_{p-q, \lambda }\left( u, v\beta +\frac{\beta }{\alpha }\sigma \right) G_{q, \lambda }\left( z\alpha + \frac{\alpha }{\beta }\rho \right) \\ & = &\sum\limits_{q = 0}^{p}\binom{p}{q}\alpha ^{q}\beta ^{p-q}\sum\limits_{\sigma = 0}^{\beta -1}\sum\limits_{\rho = 0}^{\alpha -1}(-1)^{\sigma +\rho }{}_{L}G_{p-q, \lambda }\left( u, v\alpha +\frac{\alpha }{\beta }\sigma +\rho \right) {}_{L}G_{q, \lambda }\left( z\beta +\frac{\beta }{\alpha }\rho \right) \mathit{\text{.}} \end{eqnarray*}$

Theorem 19. Let $\alpha, \beta \in \mathbb{Z}$ and $p\in \mathbb{N}_{0}$ , Then,

$\begin{eqnarray*} &&\sum\limits_{q = 0}^{p}\binom{p}{q}\beta ^{q}\alpha ^{p-q}{}_{L}G_{p-q, \lambda }(u\beta , v\beta )\sum\limits_{\sigma = 0}^{q}\binom{q}{\sigma }T_{\sigma }(\alpha -1)G_{q-\sigma , \lambda }(u\alpha ) \\ & = &\sum\limits_{q = 0}^{p}\binom{p}{q}\beta ^{p-q}\alpha ^{q}{}_{L}G_{p-q, \lambda }(u\alpha , v\alpha )\sum\limits_{\sigma = 0}^{q}\binom{q}{\sigma }T_{\sigma }(\beta -1)G_{q-\sigma , \lambda }(u\beta ). \end{eqnarray*}$

6. Concluding remark and observation

Motivated by importance and potential for applications in certain problems in number theory, combinatorics, classical and numerical analysis and other fields of applied mathematics, various special numbers and polynomials, and their variants and generalizations have been extensively investigated (for example, see the references here and those cited therein). The results presented here, being very general, can be specialized to yield a large number of identities involving known or new simpler numbers and polynomials. For example, the case $u = 0$ of the results presented here give the corresponding ones for the generalized partially degenerate Genocchi polynomials ^[3].

Acknowledgment

The authors express their thanks to the anonymous reviewers for their valuable comments and suggestions, which help to improve the paper in the current form.

Conflict of interest

We declare that we have no conflict of interests.

References

[1]	Z. A. Pawlak, Rough sets, Int. J. Comput. Inform. Sci., 11 (1982), 341–356. https://doi.org/10.1007/BF01001956 doi: 10.1007/BF01001956
[2]	Z. Pawlak, Rough concept analysis, Bulletin of the Polish Academy of Sciences Mathematics, 33 (1985), 9–10.
[3]	M. Atef, A. M. Khalil, S. G. Li, A. A. Azzam, A. E. F. El Atik, Comparison of six types of rough approximations based on j-neighborhood structure and j-adhesion neighborhood structure, J. Intell. Fuzzy Syst., 39 (2020), 4515–4531. https://doi.org/10.3233/JIFS-200482 doi: 10.3233/JIFS-200482
[4]	J. C. R. Alcantud, J. Zhan, Multi-granular soft rough covering sets, Soft Comput., 24 (2020), 9391–9402. https://doi.org/10.1007/s00500-020-04987-5 doi: 10.1007/s00500-020-04987-5
[5]	R. Mareay, R. Abu-Gdairi, M. Badr, Modeling of COVID-19 in view of rough topology, Axioms, 12 (2023), 663. https://doi.org/10.3390/axioms12070663 doi: 10.3390/axioms12070663
[6]	A. A. Azzam, A. M. Nasr, H. ElGhawalby, R. Mareay, Application on similarity relation and pretopology, Fractal Fract., 7 (2023), 168. https://doi.org/10.3390/fractalfract7020168 doi: 10.3390/fractalfract7020168
[7]	Q. Hu, L. Zhang, D. Chen, W. Pedrycz, D. Yu, Gaussian kernel based fuzzy rough sets: Model, uncertainty measures and applications, Int. J. Approx. Reason., 51 (2010), 453–471. https://doi.org/10.1016/j.ijar.2010.01.004 doi: 10.1016/j.ijar.2010.01.004
[8]	S. Boffa, B. Gerla, Sequences of refinements of rough sets: Logical and algebraic aspects, In: Transactions on Rough Sets XXII, Springer, Berlin, Heidelberg, 12485 (2020), 26–122. https://doi.org/10.1007/978-3-662-62798-3_3
[9]	S. Boffa, B. Gerla, Kleene algebras as sequences of orthopairs, In: Advances in Intelligent Systems and Computing, Springer, Cham, 641 (2018). https://doi.org/10.1007/978-3-319-66830-7_22
[10]	Y. Y. Yao, Relational interpretations of neighborhood operators and rough set approximation operators, Inform. Sciences, 111 (1998), 239–259. https://doi.org/10.1016/S0020-0255(98)10006-3 doi: 10.1016/S0020-0255(98)10006-3
[11]	Y. Yao, B. Yao, Covering based rough set approximations, Inform. Sciences, 200 (2012), 91–107. https://doi.org/10.1016/j.ins.2012.02.065 doi: 10.1016/j.ins.2012.02.065
[12]	I. Couso, D. Dubois, Rough sets, coverings and incomplete information, Fund. Inform., 108 (2011), 223–247. https://doi.org/10.3233/FI-2011-421 doi: 10.3233/FI-2011-421
[13]	Z. Bonikowski, E. Bryniarski, U. Wybraniec-Skardowska, Extensions and intentions in the rough set theory, Inform. Sciences, 107 (1998), 149–167. https://doi.org/10.1016/S0020-0255(97)10046-9 doi: 10.1016/S0020-0255(97)10046-9
[14]	W. Zhu, F. Y. Wang, On three types of covering-based rough sets, IEEE T. Knowl. Data En., 19 (2007), 1131–1144. https://doi.org/10.1109/TKDE.2007.1044 doi: 10.1109/TKDE.2007.1044
[15]	W. Zhu, F. Y. Wang, The fourth type of covering-based rough sets, Inform. Sciences, 201 (2012), 80–92. https://doi.org/10.1016/j.ins.2012.01.026 doi: 10.1016/j.ins.2012.01.026
[16]	E. C. C. Tsang, C. Degang, D. S. Yeung, Approximations and reducts with covering generalized rough sets, Comput. Math. Appl., 56 (2008), 279–289. https://doi.org/10.1016/j.camwa.2006.12.104 doi: 10.1016/j.camwa.2006.12.104
[17]	R. Mareay, I. Noaman, R. Abu-Gdairi, M. Badr, On covering-based rough intuitionistic fuzzy sets, Mathematics, 10 (2022), 4079. https://doi.org/10.3390/math10214079 doi: 10.3390/math10214079
[18]	X. Song, G. Liu, J. Liu, The relationship between coverings and tolerance relations, Int. J. Granul. Comput. Rough Set. Intell. Syst., 2010,343–354. https://doi.org/10.1504/IJGCRSIS.2010.036977 doi: 10.1504/IJGCRSIS.2010.036977
[19]	D. Dubois, H. Prade, Rough fuzzy sets and fuzzy rough sets, Int. J. Gen. Syst., 17 (1990), 191–209. https://doi.org/10.1080/03081079008935107 doi: 10.1080/03081079008935107
[20]	T. Feng, S. P. Zhang, J. S. Mi, The reduction and fusion of fuzzy covering systems based on the evidence theory, Int. J. Approx. Reason., 53 (2012), 87–103. https://doi.org/10.1016/j.ijar.2011.10.002 doi: 10.1016/j.ijar.2011.10.002
[21]	T. J. Li, Y. Leung, W. X. Zhang, Generalized fuzzy rough approximation operators based on fuzzy coverings, Int. J. Approx. Reason., 48 (2008), 836–856. https://doi.org/10.1016/j.ijar.2008.01.006 doi: 10.1016/j.ijar.2008.01.006
[22]	T. Deng, Y. Chen, W. Xu, Q. Dai, A novel approach to fuzzy rough sets based on a fuzzy covering, Inform. Sciences, 177 (2007), 2308–2326. https://doi.org/10.1016/j.ins.2006.11.013 doi: 10.1016/j.ins.2006.11.013
[23]	L. Ma, Two fuzzy covering rough set models and their generalizations over fuzzy lattices, Fuzzy Set. Syst., 294 (2016), 1–17. https://doi.org/10.1016/j.fss.2015.05.002 doi: 10.1016/j.fss.2015.05.002
[24]	B. Yang, B. Q. Hu, On some types of fuzzy covering-based rough sets, Fuzzy Set. Syst., 312 (2017), 36–65. https://doi.org/10.1016/j.fss.2016.10.009 doi: 10.1016/j.fss.2016.10.009
[25]	A. A. El-Atik, R. Abu-Gdairi, A. A. Nasef, S. Jafari, M. Badr, Fuzzy soft sets and decision making in ideal nutrition, Symmetry, 15 (2023), 1523. https://doi.org/10.3390/sym15081523 doi: 10.3390/sym15081523
[26]	B. Yang, B. Q. Hu, Fuzzy neighborhood operators and derived fuzzy coverings, Fuzzy Set. Syst., 370 (2019), 1–33. https://doi.org/10.1016/j.fss.2018.05.017 doi: 10.1016/j.fss.2018.05.017
[27]	L. Deer, C. Cornelis, L. Godo, Fuzzy neighborhood operators based on fuzzy coverings, Fuzzy Set. Syst., 312 (2107), 17–35. https://doi.org/10.1016/j.fss.2016.04.003 doi: 10.1016/j.fss.2016.04.003
[28]	D. A. Molodtsov, Soft set theory-first results, Comput. Math. Appl., 37 (1999), 19–31. https://doi.org/10.1016/S0898-1221(99)00056-5 doi: 10.1016/S0898-1221(99)00056-5
[29]	D. Molodtsov, V. Y. Leonov, D. V. Kovkov, Soft sets technique and its application, Nechetkie Sistemy i Myagkie Vychisleniya, 1 (2006), 8–39.
[30]	S. Oztunc, S. Aslan, H. Dutta, Categorical structures of soft groups, Soft Comput., 25 (2021), 3059–3064. https://doi.org/10.1007/s00500-020-05362-0 doi: 10.1007/s00500-020-05362-0
[31]	A. Mukherjee, Introduction to fuzzy sets, rough sets, and soft sets, In: Studies in Fuzziness and Soft Computing, Springer, New Delhi, 324 (2015), 1–22. https://doi.org/10.1007/978-81-322-2458-7_1
[32]	B. Sun, W. Ma, Soft fuzzy rough sets and its application in decision making, Artif. Intell. Rev., 41 (2014), 67–80. https://doi.org/10.1007/s10462-011-9298-7 doi: 10.1007/s10462-011-9298-7
[33]	J. Zhan, B. Sun, Covering-based soft fuzzy rough theory and its application to multiple criteria decision making, Comput. Appl. Math., 38 (2019), 149. https://doi.org/10.1007/s40314-019-0931-4 doi: 10.1007/s40314-019-0931-4
[34]	J. Zhan, B. Sun, On three types of soft rough covering-based fuzzy sets, J. Math., 2021 (2021), 6677298. https://doi.org/10.1155/2021/6677298 doi: 10.1155/2021/6677298
[35]	Z. Pawlak, Rough sets: Theoretical aspects of reasoning about data, Kluwer Academic Publishers, Boston, 1991.
[36]	W. Zhu, F. Y. Wang, Reduction and axiomization of covering generalized rough sets, Inform. Sciences, 152 (2003), 217–230. https://doi.org/10.1016/S0020-0255(03)00056-2 doi: 10.1016/S0020-0255(03)00056-2
[37]	F. Feng, Soft rough sets applied to multicriteria group decision making, Ann. Fuzzy Math. Inform., 2 (2011), 69–80.
[38]	F. Feng, C. Li, B. Davvaz, M. I. Ali, Soft sets combined with fuzzy sets and rough sets: A tentative approach, Soft Comput., 14 (2010), 899–911. https://doi.org/10.1007/s00500-009-0465-6 doi: 10.1007/s00500-009-0465-6

This article has been cited by:

Sarker Ashraful Islam, Farhana Kabir Esheta, Md Mahir Shahriar, Dewan Hasan Ahmed, Numerical study of aerodynamic drag reduction of a circular cylinder with an inbuilt nozzle, 2024, 1, 2996-0800, 379, 10.3934/mina.2024018

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1064) PDF downloads(79) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Tables(5)

AIMS Mathematics

Soft rough fuzzy sets based on covering

Related Papers:

Abstract

1. Introduction and preliminaries

2. Partially degenerate Laguerre-Genocchi polynomials

3. Generalized partially degenerate Laguerre-Genocchi polynomials

4. Implicit summation formulae

5. Symmetry identities

6. Concluding remark and observation

Acknowledgment

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Soft rough fuzzy sets based on covering

Related Papers:

Abstract

1. Introduction and preliminaries

2. Partially degenerate Laguerre-Genocchi polynomials

3. Generalized partially degenerate Laguerre-Genocchi polynomials

4. Implicit summation formulae

5. Symmetry identities

6. Concluding remark and observation

Acknowledgment

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog