Search

In this article, we constructed a derivative-free family of iterative techniques for extracting simultaneously all the distinct roots of a non-linear polynomial equation. Convergence analysis is discussed to show that the proposed family of iterative method has fifth order convergence. Nonlinear test models including fractional conversion, predator-prey, chemical reactor and beam designing models are included. Also many other interesting results concerning symmetric problems with application of group symmetry are also described. The simultaneous iterative scheme is applied starting with the initial estimates to get the exact roots within the given tolerance. The proposed iterative scheme requires less function evaluations and computation time as compared to existing classical methods. Dynamical planes are exhibited in CAS-MATLAB (R2011B) to show how the simultaneous iterative approach outperforms single roots finding methods that might confine the divergence zone in terms of global convergence. Furthermore, convergence domains, namely basins of attraction that are symmetrical through fractal-like edges, are analyzed using the graphical tool. Numerical results and residual graphs are presented in detail for the simultaneous iterative method. An extensive study has been made for the newly developed simultaneous iterative scheme, which is found to be efficient, robust and authentic in its domain. [ABSTRACT FROM AUTHOR]

In this paper, we provide a generating function for mix type Apostol-Genocchi polynomials of order η associated with Bell polynomials. We also derive certain important identities of Apostol Genocchi polynomials of order η based on Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some correlation with Stirling numbers and their special instances. Moreover, we discover some symmetric identities and their related known results. [ABSTRACT FROM AUTHOR]

This research article introduces a novel operator termed q-convolution, strategically integrated with foundational principles of q-calculus. Leveraging this innovative operator alongside q-Bernoulli polynomials, a distinctive class of functions emerges, characterized by both analyticity and bi-univalence. The determination of initial coefficients within the Taylor-Maclaurin series for this function class is accomplished, showcasing precise bounds. Additionally, explicit computation of the second Hankel determinant is provided. These pivotal findings, accompanied by their corollaries and implications, not only enrich but also extend previously published results. [ABSTRACT FROM AUTHOR]

The fundamental aim of this paper is to introduce the concept of poly-Jindalrae and poly-Gaenari numbers and polynomials within the context of degenerate functions. Furthermore, we give explicit expressions for these polynomial sequences and establish combinatorial identities that incorporate these polynomials. This includes the derivation of Dobinski-like formulas, recurrence relations, and other related aspects. Additionally, we present novel explicit expressions and identities of unipoly polynomials that are closely linked to some special numbers and polynomials. [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]

In mathematics, physics, and engineering, orthogonal polynomials and special functions play a vital role in the development of numerical and analytical approaches. This field of study has received a lot of attention in recent decades, and it is gaining traction in current fields, including computational fluid dynamics, computational probability, data assimilation, statistics, numerical analysis, and image and signal processing. In this paper, using q-Hermite polynomials, we define a new subclass of bi-univalent functions. We then obtain a number of important results such as bonds for the initial coefficients of | a 2 | , | a 3 | , and | a 4 | , results related to Fekete–Szegö functional, and the upper bounds of the second Hankel determinant for our defined functions class. [ABSTRACT FROM AUTHOR]

In this paper, we consider fully degenerate Daehee numbers and polynomials by using degenerate logarithm function. We investigate some properties of these numbers and polynomials. We also introduce higher-order multiple fully degenerate Daehee polynomials and numbers which can be represented in terms of Riemann integrals on the interval 0 , 1 . Finally, we derive their summation formulae. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new operator in order to derive some new symmetric properties of k -Fibonacci and k -Lucas numbers and Fibonacci polynomials. By making use of the new operator defined in this paper, we give some new generating functions for k -Fibonacci and Pell numbers and Fibonacci polynomials. [ABSTRACT FROM AUTHOR]

The present article is aimed at introducing and investigating a new class of q -hybrid special polynomials, namely, q -Fubini-Appell polynomials. The generating functions, series representations, and certain other significant relations and identities of this class are established. Some members of q -Fubini-Appell polynomial family are investigated, and some properties of these members are obtained. Further, the class of 3-variable q -Fubini-Appell polynomials is also introduced, and some formulae related to this class are obtained. In addition, the determinant representations for these classes are established. [ABSTRACT FROM AUTHOR]

This paper gives an overview of systematic and analytic approach of operational technique involves to study multi-variable special functions significant in both mathematical and applied framework and to introduce new families of special polynomials. Motivation of this paper is to construct a new class of generalized Fubini-type polynomials of the parametric kind via operational view point. The generating functions, differential equations, and other properties for these polynomials are established within the context of the monomiality principle. Using the generating functions, various interesting identities and relations related to the generalized Fubini-type polynomials are derived. Further, we obtain certain partial derivative formulas including the generalized Fubini-type polynomials. In addition, certain members belonging to the aforementioned general class of polynomials are considered. The numerical results to calculate the zeros and approximate solutions of these polynomials are given and their graphical representation are shown. [ABSTRACT FROM AUTHOR]

Kim et al. (Proc. Jangjeon Math. Soc. 21(4):589–598, 2018) have studied the central Fubini polynomials associated with central factorial numbers of the second kind. Motivated by their work, we introduce degenerate version of the central Fubini polynomials. We show that these polynomials can be represented by the fermionic p-adic integral on Z p . From the fermionic p-adic integral equations, we derive some new properties related to degenerate central factorial numbers of the second kind and degenerate Euler numbers of the second kind. [ABSTRACT FROM AUTHOR]

In this article, the general polynomials are taken as base with the Gould-Hopper matrix polynomials to introduce a family of 3-variable general-Gould-Hopper matrix polynomials (3VgGHMaP). These polynomials are framed within the context of monomiality principle and their properties are established. Examples of some members belonging to this family are considered. Certain bilateral and bilinear generating matrix functions for 3Vg-GHMaP are also derived. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce a new family of polynomials, which are called the truncated-exponential based Frobenius–Euler polynomials, based upon an exponential generating function. By making use of this exponential generating function, we obtain their several new properties and explicit summation formulas. Finally, we consider the truncated-exponential based Apostol-type Frobenius–Euler polynomials and their quasi-monomial properties. [ABSTRACT FROM AUTHOR]

We consider a new class of generating functions of the generalizations of Bernoulli and Euler polynomials in terms of (p, q)-integers. By making use of these generating functions, we derive (p, q)-generalizations of several old and new identities concerning Apostol-Bernoulli and Apostol-Euler polynomials. Finally, we define the (p, q)-generalization of Stirling polynomials of the second kind of order v, and provide a link between the (p, q)-generalization of Bernoulli polynomials of order v and the (p, q)-generalization of Stirling polynomials of the second kind of order v. [ABSTRACT FROM AUTHOR]

Let F, N, A and N2 denote the properties of being finite, nilpotent, abelian and nilpotent of classes at most 2, respectively. Firstly we consider the class of finitely generated FN-groups. We show that the property FC is closed under finite extensions, and extend this result to finitely generated NF-groups. Secondly we prove that a finitely generated NF-group G is in the class ((FC)F, ∞) if and only if G is an FA-group. Finally we prove that a finitely generated NF-group in the class ((FC)F, ∞)* is an FN2-group. Moreover, G/Z2(G) is finite. [ABSTRACT FROM AUTHOR]

The present paper deals with the approximation properties of the univariate operators which are the generalization of the Kantorovich-Szász type operators involving Brenke type polynomials. We investigate the order of convergence by using Peetre's K-functional and the Ditzian-Totik modulus of smoothness and study the degree of approximation of the univariate operators for continuous functions in a Lipschitz space, a Lipschitz type maximal function and a weighted space. The rate of approximation of functions having derivatives equivalent with a function of bounded variation is also obtained. [ABSTRACT FROM AUTHOR]

In the present paper, we define a sequence of bivariate operators by linking the Bernstein-Chlodowsky operators and the Sz'asz-Kantorovich operators based on Appell polynomials. First, we establish the moments of the operators and then determine the rate of convergence of these operators in terms of the total and partial modulus of continuity. Next, we obtain the order of approximation of the considered operators in a weighted space. Furthermore, we define the associated GBS (Generalized Boolean Sum) operators of the linking operators and then study the rate of convergence with the aid of the Lipschitz class of Bögel continuous functions and the mixed modulus of smoothness. [ABSTRACT FROM AUTHOR]

The present paper introduces the Szász-Durrmeyer type operators based on Boas-Buck type polynomials which include Brenke type polynomials, Sheffer polynomials and Appell polynomials considered by Sucu et al. (Abstr. Appl. Anal. 2012:680340, 2012). We establish the moments of the operator and a Voronvskaja type asymptotic theorem and then proceed to studying the convergence of the operators with the help of Lipschitz type space and weighted modulus of continuity. Next, we obtain a direct approximation theorem with the aid of unified Ditzian-Totik modulus of smoothness. Furthermore, we study the approximation of functions whose derivatives are locally of bounded variation. [ABSTRACT FROM AUTHOR]

In the paper, we introduce an analogue of Haar distribution based on ( ρ , q ) -numbers, as follows: μ ρ , q ( a + p N Z p ) = ρ p N [ p N ] ρ , q ( q ρ ) a . By means of this distribution, we derive ( ρ , q ) -analogue of Volkenborn integration which is a new generalization of Kim's q -Volkenborn integration defined in [11] . From this definition, we investigate some properties of Volkenborn integration based on ( ρ , q ) -numbers. Finally, we construct ( ρ , q ) -Bernoulli numbers and polynomials derived from ( ρ , q ) -Volkenborn integral and obtain some their properties. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generating function for a new generalization of Laguerre-based Apostol-Bernoulli polynomials, Apostol-Euler and Apostol-Genocchi polynomials. By making use of the generating function method and some functional equations mentioned in the paper, we conduct a further investigation in order to obtain symmetric identities of these polynomials. [ABSTRACT FROM AUTHOR]

By using the modified Milne-Thomson's polynomial given in Araci et al. (Appl Math Inf Sci 8(6):2803-2808, ), we introduce a new concept of the Apostol Hermite-Genocchi polynomials. We also perform a further investigation for aforementioned polynomial and derive some implicit summation formulae and general symmetric identities arising from different analytical means and generating functions method. The results obtained here are an extension of Hermite-Bernoulli polynomials (Pathan and Khan in Mediterr J Math 12:679-695, ) and Hermite-Euler polynomials (Pathan and Khan in Mediterr J Math , doi:) to Apostol type Hermite-Genocchi polynomials defined in this paper. [ABSTRACT FROM AUTHOR]

We introduce Dirichlet's type of twisted Eulerian polynomials by using p-adic fermionic q-invairant integral in the p-adic integer ring and obtain some new interesting identities. Using a complex contour integral representation based on the generating function of Dirichlet's type of twisted Eulerian polynomials, we get L-type function which interpolates for Dirichlet's type of Eulerian polynomials at negative integers. [ABSTRACT FROM AUTHOR]

The Newton power-sum formulas relate to sums of powers of roots of a polynomial with the coefficients of the polynomial. In this paper we obtain formulas that relate to sums of reciprocal powers of zeros and poles of entire and meromorphic functions with the coefficients of their Taylor series expansions. We then derive the recurrence formulas for the Riemann zeta function at integer arguments and compute the sums extended over the nontrivial zeros of the Riemann zeta function. [ABSTRACT FROM AUTHOR]

Recently, the von Staudt–Clausen’s theorem for q -Euler numbers was introduced by Kim (2013) and q -Genocchi numbers were constructed by Araci et al. (2013). In this paper, we give the corresponding von Staudt–Clausen’s theorem for q -Genocchi numbers and also get the Kummer-type congruence for q -Genocchi numbers. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to deal with applications of umbral calculus on fermionic p-adic integral on . From those applications, we derive some new identities on Genocchi numbers and polynomials. Moreover, a systemic study of the class of Sheffer sequences on the generating function of Genocchi polynomials are given. [Copyright &y& Elsevier]

In the present paper, we introduce the new generalization of q-Genocchi polynomials and numbers of higher order. Also, we give some interesting identities. Finally, by applying q-Mellin transformation to the generating function for q-Genocchi polynomials of higher order put we define novel q-Hurwitz-Zeta type function which is an interpolation for this polynomials at negative integers. [ABSTRACT FROM AUTHOR]

We consider the following boundary-value problem of nonlinear fractional differential equation with p-Laplacian operator Dβ0+(ɸp(Dα0+u(t))) + a(t)f(u) = 0,0 < t < 1, u(0) = yu(h) + λ, u'(0) = u, ɸp(Dα0+u(0))))'' = ɸp(Dα0+u(0))))''' = 0, where 1 < α ≤ 2, 3 < β ≤ 4 are real numbers, Dα0+, Dβ0+ are the standard Caputo fractional derivatives, ɸp(s) = |s|p-2s, P > 1, ɸ-1p = ɸq,1/P + 1/q = 1, 0 ≤ y < 1, 0 ≤ h ≤ 1, λ, μ > 0 are parameters, a : (0, 1) → [0, +8), and f : [0, +8) → [0, +8) are continuous. By the properties of Green function and Schauder fixed point theorem, several existence and nonexistence results for positive solutions, in terms of the parameters λ and μ are obtained. The uniqueness of positive solution on the parameters λ and μ is also studied. In the final section of this paper, we derive not only new but also interesting identities related special polynomials by which Caputo fractional derivative. [ABSTRACT FROM AUTHOR]

The essential aim of this paper is to introduce novel identities for q-Genocchi numbers and polynomials by using themethod by T. Kim et al. (article in press).We show that these polynomials are related to p-adic analogue of Bernstein polynomials. Also, we derive relations between q-Genocchi and q-Bernoulli numbers. [ABSTRACT FROM AUTHOR]

We deal with q-Euler numbers and q-Bernoulli numbers. We derive some interesting relations for q- Euler numbers and polynomials by using their generating function and derivative operator. Also, we derive relations between the q-Euler numbers and q-Bernoulli numbers via the p-adic q-integral in the p-adic integer ring. [ABSTRACT FROM AUTHOR]

The fundamental aim of this paper is to construct (h, q)-Genocchi numbers and polynomials with weight α.We shall obtain some interesting relations by using p-adic q-integral on ℤp; in the sense of fermionic. Also, we shall derive the (h, q)-extensions of zeta type functions with weight α from the (h, q) a Mellin transformation of this generating function which interpolates the (h, q)-Genocchi numbers and polynomials with weight α at negative integers. [ABSTRACT FROM AUTHOR]

In this paper, we consider Bell-based Stirling polynomials of the second kind and derive some useful relations and properties including some summation formulas related to the Bell polynomials and Stirling numbers of the second kind. Then, we introduce Bell-based Bernoulli polynomials of order α and investigate multifarious correlations and formulas including some summation formulas and derivative properties. Also, we acquire diverse implicit summation formulas and symmetric identities for Bell-based Bernoulli polynomials of order α. Moreover, we attain several interesting formulas of Bell-based Bernoulli polynomials of order α arising from umbral calculus. [ABSTRACT FROM AUTHOR]

In this paper, we employ an umbral method to reformulate the 3-variable Hermite polynomials and introduce the 4-parameter 3-variable Hermite polynomials. We also obtain some new properties for these polynomials. Moreover, some special cases are discussed and some concluding remarks are also given. [ABSTRACT FROM AUTHOR]

Recently, Kim and Kim (Russ. J. Math. Phys. 27(2):227–235, 2020) have studied new type degenerate Bernoulli numbers and polynomials by making use of degenerate logarithm. Motivated by (Kim and Kim in Russ. J. Math. Phys. 27(2):227–235, 2020), we consider a special class of polynomials, which we call a new type of degenerate Daehee numbers and polynomials of the second kind. By using their generating function, we derive some new relations including the degenerate Stirling numbers of the first and second kinds. Moreover, we introduce a new type of higher-order degenerate Daehee polynomials of the second kind. We also derive some new identities and properties of this type of polynomials. [ABSTRACT FROM AUTHOR]

Kim and Kim (Russ. J. Math. Phys. 26(1):40–49, 2019) have studied the type 2 poly-Bernoulli polynomials. Inspired by their work, we consider a new class of the Frobenius–Genocchi polynomials, which is called the type 2 poly-Frobenius–Genocchi polynomials, by means of the polyexponential function. We also derive some new relations and properties including the Stirling numbers of the first and second kinds. In a special case, we give a relation between the type 2 poly-Frobenius–Genocchi polynomials and Bernoulli polynomials of order k. Moreover, motivated by the definition of the unipoly-Bernoulli polynomials given in (Kim and Kim in Russ. J. Math. Phys. 26(1):40–49, 2019), we introduce the unipoly-Frobenius–Genocchi polynomials via a unipoly function and give multifarious properties including derivative and integral properties. Furthermore, we provide a correlation between the unipoly-Frobenius–Genocchi polynomials and the classical Frobenius–Genocchi polynomials. [ABSTRACT FROM AUTHOR]

In this paper, the class of the twice-iterated 2D q-Appell polynomials is introduced. The generating function, series definition and some relations including the recurrence relations and partial q-difference equations of this polynomial class are established. The determinant expression for the twice-iterated 2D q-Appell polynomials is also derived. Further, certain twice-iterated 2D q-Appell and mixed type special q-polynomials are considered as members of this polynomial class. The determinant expressions and some other properties of these associated members are also obtained. The graphs and surface plots of some twice-iterated 2D q-Appell and mixed type 2D q-Appell polynomials are presented for different values of indices by using Matlab. Moreover, some areas of potential applications of the subject matter of, and the results derived in, this paper are indicated. [ABSTRACT FROM AUTHOR]

In this paper, we propose to investigate the truncated-exponential-based Apostol-type polynomials and derive their various properties. In particular, we establish the operational correspondence between this new family of polynomials and the familiar Apostol-type polynomials. We also obtain some implicit summation formulas and symmetric identities by using their generating functions. The results, which we have derived here, provide generalizations of the corresponding known formulas including identities involving generalized Hermite-Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

In this paper, the class of q -Sheffer–Appell polynomials is introduced. The generating function, series definition, determinant definition and some other identities of this class are established. Certain members of q -Sheffer–Appell polynomials are investigated and some properties of these members are derived. In addition, the class of 2D q -Sheffer–Appell polynomials is introduced. Further, the graphs of some members of q -Sheffer–Appell polynomials and 2D q -Sheffer–Appell polynomials are plotted for different values of indices by using Matlab. [ABSTRACT FROM AUTHOR]

In this paper, we present a systematic and unified investigation for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. By applying the generating-function methods and summation-transform techniques, we establish some higher-order convolutions for the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials and the Apostol-Genocchi polynomials. Some results presented here are the corresponding extensions of several known formulas. [ABSTRACT FROM AUTHOR]

The article is written with the objectives to introduce a multi-variable hybrid class, namely the Hermite–Apostol-type Frobenius–Euler polynomials, and to characterize their properties via different generating function techniques. Several explicit relations involving Hurwitz–Lerch Zeta functions and some summation formulae related to these polynomials are derived. Further, we establish certain symmetry identities involving generalized power sums and Hurwitz–Lerch Zeta functions. An operational view for these polynomials is presented, and corresponding applications are given. The illustrative special cases are also mentioned along with their generating equations. [ABSTRACT FROM AUTHOR]

The significance of multi-variable special polynomials has been identified both in mathematical and applied frameworks. The article aims to focus on a new class of 3-variable Legendre-truncated-exponential-based Sheffer sequences and to investigate their properties by means of Riordan array techniques. The quasi-monomiality of these sequences is studied within the context of Riordan arrays. These sequences are expressed in determinant forms by utilizing the relation between the Sheffer sequences and Riordan arrays. Certain special Sheffer sequences are used to construct the Legendre-truncated-exponential-based Chebyshev polynomials. The same polynomials are used as base to introduce the hybrid classes involving the exponential polynomials and the Euler polynomials of higher order as illustrative examples of the aforementioned general class of polynomials. The shapes are shown and zeros are computed for these sequences by using mathematical software. The main purpose is to demonstrate the advantage of using numerical investigation and computations to discover fascinating pattern of scattering of zeros through graphical representations. [ABSTRACT FROM AUTHOR]