Recently, D. Zhang et al. introduced the generalized Choquet integral, extending pseudo-integrals and Choquet-like integrals while exploring their foundational properties. Building on this framework, we introduce the concept of generalized Choquet integrals for triangular fuzzy number (TFN)-valued functions, referred to as TGC-integrals. This work investigates the key properties of TGC-integrals, including monotone non-decreasing convergence theorems and inequalities such as the Fatou type, Jensen type, Minkowski type, and Holder type inequalities, specifically tailored for TFN-valued functions. Furthermore, we provide illustrative examples that demonstrate practical applications of TGC-integrals, such as TFN-valued Choquet expected utility and pseudo-functional analysis. These results establish a robust theoretical foundation for analyzing TFN-valued functions and highlight their potential for addressing uncertainty and ambiguity in real-world problems.

This study introduces the Choquet integral of fuzzifying functions with respect to a fuzzy measure. To express various phenomena or ambiguous values in many applications may not be enough to show them as a function, in which case a fuzzifying function can be applied to achieve better expressions or flexibility for given function values. To apply Choquet integrals of fuzzifying functions, we consider Choquet integrals of interval-valued functions as an operator which are alpha-level functions of fuzzifying functions. In this study, we investigate some properties of Choquet integral of fuzzifying functions and present their applications. As part of this, a series of relevant examples and their subsequent applications are provided, along with the fuzzification of two integrands: A probability density function (PDF) and a utility function.

A new family of p-Bernoulli numbers and polynomials was introduced by Rahmani (J. Number Theory 157:350–366, 2015) with the help of the Gauss hypergeometric function. Motivated by that paper and in the light of the recent interests in finding degenerate versions, we construct the generalized degenerate Bernoulli numbers and polynomials by means of the Gauss hypergeometric function. In addition, we construct the degenerate type Eulerian numbers as a degenerate version of Eulerian numbers. For the generalized degenerate Bernoulli numbers, we express them in terms of the degenerate Stirling numbers of the second kind, of the degenerate type Eulerian numbers, of the degenerate p-Stirling numbers of the second kind and of an integral on the unit interval. As to the generalized degenerate Bernoulli polynomials, we represent them in terms of the degenerate Stirling polynomials of the second kind.

Recently, Kim-Kim (2019) introduced polyexponential and unipoly functions. By using these functions, they defined type 2 poly-Bernoulli and type 2 unipoly-Bernoulli polynomials and obtained some interesting properties of them. Motivated by the latter, in this paper, we construct the poly-Genocchi polynomials and derive various properties of them. Furthermore, we define unipoly Genocchi polynomials attached to an arithmetic function and investigate some identities of them.

Recently, degenerate polylogarithm functions were introduced by Kim and Kim. In this paper, we introduce degenerate poly-Bernoulli polynomials by means of the degenerate polylogarithm functions and investigate some their properties. In more detail, we find certain explicit expressions for those polynomials in terms of the Carlitz degenerate Bernoulli polynomials and the degenerate Stirling numbers of the second kind. Furthermore, we obtain some expressions for differences of the degenerate poly-Bernoulli polynomials.

Wang-Qu [9]는 패턴 인식, 이미지 처리, 근사 추론, 퍼지 제어 등 다양한 분야에서 광범위하게 적용할 수 있는 애매한 소프트 집합의 엔트로피, 유사성 측도 및 거리 측도를 도입했다. 또한, Jang-Kwon [3]에서 처음으로 구간값 함수의 쇼케이 적분을 정의하고 이와 관련된 성질을 조사하였다. 그이후 많은 논문에서 구간값 쇼케이적분의 응용을 연구해 왔다. 애매한 집합의 속성이 구간값 소속함수의 성질을 가지고 있음을 이용하고자 한다. 본 논문에서는 애매한 소프트 집합상에서 쇼케이 적분을 정의하고, 이들 적분에 의해 정의된 구간값 거리 측도를 조사한다.

The aim of this paper is to study Jindalrae and Gaenari numbers and polynomials in connection with Jindalrae–Stirling numbers of the first and second kinds. For this purpose, we first introduce Jindalrae–Stirling numbers of the first and second kinds as extensions of the notions of the degenerate Stirling numbers of the first and second kinds, and deduce several relations involving those special numbers. Then we introduce Jindalrae and Gaenari numbers and polynomials and obtain some explicit expressions and identities associated with those numbers and polynomials. In addition, we interpret our results by using umbral calculus.

The aim of this paper is to study Jindalrae and Gaenari numbers and polynomials in connection with Jindalrae-Stirling numbers of the first and second kinds. For this purpose, we first introduce Jindalrae-Stirling numbers of the first and second kinds as extensions of the notions of the degenerate Stirling numbers of the first and second kinds, and deduce several relations involving those special numbers. Then we introduce Jindalrae and Gaenari numbers and polynomials and obtain some explicit expressions and identities associated with those numbers and polynomials. In addition, we interpret our results by using umbral calculus.

We consider the modified degenerate q-Daehee polynomials and numbers of the second kind which can be represented as the p-adic q-integral. Furthermore, we investigate some properties of those polynomials and numbers.

In this paper, we study Carlitz's type q-Daehee polynomials and investigate the symmetric identities for them by using the p-adic q-integral on Zp under the symmetry group of degree n.

Recently, Masjed-Jamei, Beyki, and Koepf studied the so-called new type Euler polynomials without using Euler polynomials of complex variable. Here we study the type 2 degenerate cosine-Euler and type 2 degenerate sine-Euler polynomials, which are type 2 degenerate versions of these new type Euler polynomials, by considering the degenerate Euler polynomials of complex variable and by treating the real and imaginary parts separately. In addition, we investigate the corresponding ones for Bernoulli polynomials in the same manner. We derive some explicit expressions for those new polynomials and some identities relating to them. Here we note that the idea of separating the real and imaginary parts separately gives an affirmative answer to the question asked by Hacène Belbachir.

Recently, extended r-central factorial numbers of the second kind and extended r-central Bell polynomials were introduced and various results of them were investigated. The purpose of this paper is to further derive properties, recurrence relations and identities related to these numbers and polynomials using umbral calculus techniques. Especially, we will represent the extended r-central Bell polynomials in terms of quite a few families of well-known special polynomials.

Recently, Dolgy-Jang-Kwon-Kim introduced Carlitz’s type q-Changhee polynomials. In this paper, we define Carlitz’s type modified degenerate q-Changhee polynomials and investigate some interesting identities of these polynomials.

In the paper a method is proposed for evaluation of the students' knowledge obtained in the university e-learning courses and an evaluation of the whole student class. For the assessment of the student's solution of the respective assessment units the theory of intuitionistic fuzzy sets is used, while for the class evaluation, interval valued intuitionistic fuzzy sets is used. The obtained intuitionistic fuzzy estimations reflect the degree of each student's good or poor performances, for each assessment unit. The interval valued intuitionistic fuzzy evaluations are based on the separate student's evaluations. We also consider a degree of uncertainty that represents such cases wherein the student is currently unable to solve the problem. The method presented here provides the possibility for the algorithmization of the process of forming the student's evaluations.

In this paper, we consider the modified partially degenerate Genocchi polynomials and investigate some properties of these polynomials. From these properties, we give some new and interesting identities of them.

In this paper, we give some identities of lambda-Daehee polynomials and investigate a new and interesting identities of lambda-Daehee polynomial arising from the symmetry properties of the p-adic invariant integral on Zp. (C) 2017 All rights reserved.

Jeong-Rim-Kim(2015) studied the degenerate Cauchy numbers and polynomials and investigated some properties of these k-times degenerate Cauchy numbers and polynomials. In this paper, we define the degenerate Genocchi polynomials and the k-th degeneration of Genocchi polynomials, and investigate some properties of these polynomials

By using the Bosonic p-adic integral, Kim et al. [D. S. Kim, T. Kim, H.-I. Kwon, J.-J. Seo, Adv. Stud. Theor. Phys., 8 (2014), 745-754] studied some identities of the Korobov and Daehee mixed-type polynomials. In this paper, by using the fermionic p-adic integral, we define the Korobov and Changhee mixed-type polynomials and give some interesting identities of those polynomials. (C) 2017 All rights reserved.

In this paper, we study some properties of degenerate Changhee-Genocchi numbers and polynomials and give some new identities of these polynomials and numbers which are derived from the generating function. In particular, we provide interesting identities related to the Changhee-Genocchi polynomials of the second kind and Changhee-Genocchi numbers of the second kind.

In this paper we consider differential equations which are closely related to the generating functions of Euler numbers. By using the same method of Kim's calculation in Kim [24,25], we derive identities involving Euler numbers arising from differential equations. In particular, we derive some new identities between the sums of Euler numbers and Genocchi numbers of higher order.