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In this paper, by using the Golden Calculus, we introduce the generalized Apostol-type Frobenius–Euler–Fibonacci polynomials and numbers; additionally, we obtain various fundamental identities and properties associated with these polynomials and numbers, such as summation theorems, difference equations, derivative properties, recurrence relations, and more. Subsequently, we present summation formulas, Stirling–Fibonacci numbers of the second kind, and relationships for these polynomials and numbers. Finally, we define the new family of the generalized Apostol-type Frobenius–Euler–Fibonacci matrix and obtain some factorizations of this newly established matrix. Using Mathematica, the computational formulae and graphical representation for the mentioned polynomials are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we generate new degenerate quantum Euler polynomials (DQE polynomials), which are related to both degenerate Euler polynomials and q-Euler polynomials. We obtain several (q , h) -differential equations for DQE polynomials and find some relations of q-differential and h-differential equations. By varying the values of q , η , and h, we observe the values of DQE numbers and approximate roots of DQE polynomials to obtain some properties and conjectures. [ABSTRACT FROM AUTHOR]

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to organize various types of higher order q-differential equations that are connected to q-sigmoid polynomials and obtain certain properties regarding their solutions. Using the properties of q-sigmoid polynomials, we show the symmetric properties of q-differential equations of higher order. Moreover, we derive special properties for the approximate roots of q-sigmoid polynomials which are solutions of higher order q-differential equations. [ABSTRACT FROM AUTHOR]

Utilizing p , q -numbers and p , q -concepts, in 2016, Duran et al. considered p , q -Genocchi numbers and polynomials, p , q -Bernoulli numbers and polynomials and p , q -Euler polynomials and numbers and provided multifarious formulas and properties for these polynomials. Inspired and motivated by this consideration, many authors have introduced (p , q) -special polynomials and numbers and have described some of their properties and applications. In this paper, using the (p , q) -cosine polynomials and (p , q) -sine polynomials, we consider a novel kinds of (p , q) -extensions of geometric polynomials and acquire several properties and identities by making use of some series manipulation methods. Furthermore, we compute the p , q -integral representations and p , q -derivative operator rules for the new polynomials. Additionally, we determine the movements of the approximate zerosof the two mentioned polynomials in a complex plane, utilizing the Newton method, and we illustrate them using figures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new class of Bell-based Apostol-type Frobenius–Euler polynomials and investigate some properties of these polynomials. We derive some explicit and implicit summation formulas and their symmetric identities by using different analytical means and applying generating functions of generalized Apostol-type Frobenius-Euler polynomials and Bell-based Apostol-type Frobenius-Euler polynomials. In particular, parametric kinds of the Bell-based Apostol-type Frobenius-Euler polynomials are introduced and some of their algebraic and analytical properties are established. In addition, illustrative examples of these families of polynomials are shown, focusing on their numerical values and piloting some beautiful computer-aided graphs of them. [ABSTRACT FROM AUTHOR]

One finds several q-differential equations of a higher order for q-Euler polynomials and q-Genocchi polynomials. Additionally, we have a few q-differential equations of a higher order, which are mixed with q-Euler numbers and q-Genocchi polynomials. Moreover, we investigate some symmetric q-differential equations of a higher order by applying symmetric properties of q-Euler polynomials and q-Genocchi polynomials. [ABSTRACT FROM AUTHOR]

Hermite polynomials are one of the Apell polynomials and various results were found by the researchers. Using Hermit polynomials combined with q-numbers, we derive different types of differential equations and study these equations. From these equations, we investigate some identities and properties of q-Hermite polynomials. We also find the position of the roots of these polynomials under certain conditions and their stacked structures. Furthermore, we locate the roots of various forms of q-Hermite polynomials according to the conditions of q-numbers, and look for values which have approximate roots that are real numbers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially degenerate Hermite polynomials. Finally, we study the symmetric properties of the structure of the roots of the two-variable partially degenerate Hermite equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in a systematic way. Furthermore, we consider some relationships for parametric kind of Fubini polynomials associated with Bernoulli, Euler, and Genocchi polynomials and Stirling numbers of the second kind. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials and study their basic properties. Finally, we consider some relationships for Stirling numbers of the second kind related to bivariate kinds of poly-Bernoulli and poly-Genocchi polynomials. [ABSTRACT FROM AUTHOR]

The intended objective of this study is to define and investigate a new class of q-generalized tangent-based Appell polynomials by combining the families of 2-variable q-generalized tangent polynomials and q-Appell polynomials. The investigation includes derivations of generating functions, series definitions, and several important properties and identities of the hybrid q-special polynomials. Further, the analogous study for the members of this q-hybrid family are illustrated. The graphical representation of its members is shown, and the distributions of zeros are displayed. [ABSTRACT FROM AUTHOR]

In this article, we introduce the parametric kinds of degenerate type Fubini polynomials and numbers. We derive recurrence relations, identities and summation formulas of these polynomials with the aid of generating functions and trigonometric functions. Further, we show that the parametric kind of the degenerate type Fubini polynomials are represented in terms of the Stirling numbers. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the two variable degenerate Hermite equations. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study some interesting identities for the multiple twisted (p , q) -L-function in a complex field. First, we construct new generating functions of the new Carlitz-type higher order twisted (p , q) -Euler numbers and polynomials. By applying the Mellin transformation to these generating functions, we obtain integral representations of the multiple twisted (p , q) -Euler zeta function and multiple twisted (p , q) -L-function, which interpolate the Carlitz-type higher order twisted (p , q) -Euler numbers and Carlitz-type higher order twisted (p , q) -Euler polynomials at non-positive integers, respectively. Second, we get some explicit formulas and properties, which are related to Carlitz-type higher order twisted (p , q) -Euler numbers and polynomials. Third, we give some new symmetric identities for the multiple twisted (p , q) -L-function. Furthermore, we also obtain symmetric identities for Carlitz-type higher order twisted (p , q) -Euler numbers and polynomials by using the symmetric property for the multiple twisted (p , q) -L-function. [ABSTRACT FROM AUTHOR]

In this paper, we study differential equations arising from the generating function of the (r , β) -Bell polynomials. We give explicit identities for the (r , β) -Bell polynomials. Finally, we find the zeros of the (r , β) -Bell equations with numerical experiments. [ABSTRACT FROM AUTHOR]

Let 1 < a < b < c < d and α ^ 5 : = 1 , a , b , c , d ∧ be a weighted sequence that is recursively generated by five weights 1 , a , b , c , d . In this paper, we give sufficient conditions for the positive quadratic hyponormalities of W α x and W α y , x , with α x : x , α ^ 5 and α y , x : y , x , α ^ 5. [ABSTRACT FROM AUTHOR]

In this paper, we study differential equations arising from the generating functions of Hermit Kamp e ´ de F e ´ riet polynomials. Use this differential equation to give explicit identities for Hermite Kamp e ´ de F e ´ riet polynomials. Finally, use the computer to view the location of the zeros of Hermite Kamp e ´ de F e ´ riet polynomials. [ABSTRACT FROM AUTHOR]