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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 present study introduces a new family of analytic functions by utilizing the q-derivative operator and the q-version of the hyperbolic tangent function. We find certain inequalities, including the coefficient bounds, second Hankel determinants, and Fekete–Szegö inequalities, for this novel family of bi-univalent functions. It is worthy of note that almost all the results are sharp, and their corresponding extremal functions are presented. In addition, some special cases are demonstrated to show the validity of our findings. [ABSTRACT FROM AUTHOR]

In this present paper, we define a new operator in conjugation with the basic (or q-) calculus. We then make use of this newly defined operator and define a new class of analytic and bi-univalent functions associated with the q-derivative operator. Furthermore, we find the initial Taylor–Maclaurin coefficients for these newly defined function classes of analytic and bi-univalent functions. We also show that these bounds are sharp. The sharp second Hankel determinant is also given for this newly defined function class. [ABSTRACT FROM AUTHOR]

The article announces the appointment of Serkan Araci as the second Editor-in-Chief of Mathematical and Computer Modelling of Dynamical Systems, joining Wolfgang Kemmetmüller. The Editorial Board has also been updated with new members from various countries, specializing in areas such as mathematical epidemiology, robotic systems, and ecological modeling. The retiring board members are thanked for their service. The article is written by Serkan Araci and Wolfgang Kemmetmüller. [Extracted from the article]

In this paper, we discuss a generalization to the Cherednik–Opdam integral operator to an abstract space of Boehmians. We introduce sets of Boehmians and establish delta sequences and certain class of convolution products. Then we prove that the extended Cherednik–Opdam integral operator is linear, bijective and continuous with respect to the convergence of the generalized spaces of Boehmians. Moreover, we derive embeddings and discuss properties of the generalized theory. Moreover, we obtain an inversion formula and provide several results. [ABSTRACT FROM AUTHOR]

This paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed. [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 aims to discuss a generalization of certain paraxial diffraction integral operator in a class of generalized functions. At the start of this paper, we propose a convolution formula and establish certain convolution theorem. Then, with the addition to the convolution theorem, we consider a set of approximating identities and substantially employ our results in generating sets of integrable and locally integrable Boehmians. The said generalized integral operator is tested and declared to be one-to-one and onto mapping. Continuity of the generalized operator with respect to the convergence of the Boehmian spaces is obtained. Over and above, an inversion formula and consistency results are also counted. [ABSTRACT FROM AUTHOR]

In this Special Issue, the recent advances in the applications of symmetric functions for mathematics and mathematical physics are reviewed, including many novel techniques in analytic functions, transformation methods, economic growth models, and Hurwitz-Lerch zeta functions that were developed to provide reliable solutions to combinatorial problems. References 1 Taj Y., Zainab S., Xin Q., Tawfiq F.M.O., Raza M., Malik S.N. Certain Coefficient Problems for q-Starlike Functions Associated with q-Analogue of Sine Function. They investigated the sharp coefficient bounds and upper bound of the third-order Hankel determinant for this class. Mondal [[2]] obtained three types of analytic functions based on their infinite product representation. [Extracted from the article]

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]

In the paper, we define Laguerre-based Hermite-Bernoulli polynomial with its generating function, and investigate certain properties. From this generating function, we derive summation formulas and related bilateral series associated with the newly introduced generating function. Some of whose special cases are also presented. Relevant connections of some results presented here with those involving simpler known partly unilateral and partly bilateral representations are also obtained. [ABSTRACT FROM AUTHOR]

In the present paper, we find the Fourier expansion of the Apostol Frobenius-Euler polynomials. By using a Fourier expansion of the Apostol Frobenius-Euler polynomials, we derive some new and interesting results. [ABSTRACT FROM AUTHOR]

Recently, a mixed hybrid operator, generalizing the well-known Phillips operators and Baskakov–Szász type operators, was introduced. In this paper, we study Bézier variant of these new operators. We investigate the degree of approximation of these operators by means of the Lipschitz class function, the modulus of continuity, and a weighted space. We study a direct approximation theorem by means of the unified Ditzian–Totik modulus of smoothness. Furthermore, the rate of convergence for functions having derivatives of bounded variation is discussed. [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]

Motivated by a number of recent investigations, we consider a new analogue of Bernstein-Durrmeyer operators based on certain variants. We derive some approximation properties of these operators. We also compute local approximation and Voronovskaja type asymptotic formula. We illustrate the convergence of aforementioned operators by making use of the software MATLAB which we stated in the paper. [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]

Recently, Sofonea (Gen. Math. 16:47-54, 2008) considered some relations in the context of quantum calculus associated with the q-derivative operator $D_{q}$ and divided difference. As applications of the post-quantum calculus known as the $(p,q)$ -calculus, we derive several relations involving the $(p,q)$ -derivative operator and divided differences. [ABSTRACT FROM AUTHOR]

In the year 2014, Kim et al. computed a kind of new sums of the products of an arbitrary number of the classical Bernoulli and Euler polynomials by using the Euler basis for the vector space of polynomials of bounded degree. Inspired by their work, in this paper, we establish some new formulas for such a kind of sums of the products of an arbitrary number of the Apostol-Bernoulli, Euler, and Genocchi polynomials by making use of the generating function methods and summation transform techniques. The results derived here are generalizations of the corresponding known formulas involving the classical Bernoulli, Euler, and Genocchi 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]

In this paper, we introduce q -Frobenius–Euler numbers and derive some new properties. From those properties, we show that this number is a p -adic integer, and can be expressed by von Staudt–Clausen's theorem. We also get a type of Kummer congruence for this number. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence of a solution arising from the following fractional q-difference boundary value problem by using the p-Laplacian operator: $D_{q}^{\gamma}(\phi_{p}(D_{q}^{\delta}y(t)))+f(t,y(t))=0$ ( $0< t<1$; $0<\gamma<1$; $3<\delta<4$), $y(0)=(D_{q}y)(0)=(D_{q}^{2}y)(0) =0$, $a_{1}(D_{q}y)(1)+a_{2}(D_{q}^{2}y)(1)=0$, $a_{1} +\vert a_{2}\vert \neq0$, $D_{0+}^{\gamma}y(t)|_{t=0}=0$. We make use of such a fractional q-difference boundary value problem in order to show the existence and uniqueness of positive and nondecreasing solutions by means of a familiar fixed point theorem. [ABSTRACT FROM AUTHOR]

In the present paper, we aim to specify a p-adic continuous function for an odd prime inside a p-adic q-analog of the extended Dedekind-type sums of higher order according to extended q-Euler polynomials (or weighted q-Euler polynomials) which is derived from a fermionic p-adic q-deformed integral on $\mathbb{Z}_{p}$. [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 this paper, we present a further investigation for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. By making use of the generating function methods and summation transform techniques, we establish some new identities involving the products of the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. Many of the results presented here are the corresponding generalizations of some known formulas on the classical Bernoulli polynomials and the classical Genocchi polynomials. [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]

In this work, we study the asymptotic properties of a new Sturm-Liouville problem with retarded argument. Contrary to previous works, differential equation includes eigenparameter as a quadratic function. In the considered problem arise new difficulties. Copyright © 2013 John Wiley & Sons, Ltd. [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 give some new formulae for product of two Genocchi polynomials including Euler polynomials and Bernoulli polynomials. Moreover, we derive some applications for Genocchi polynomials to study a matrix formulation. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to derive some applications of umbral calculus by using extended fermionic p-adic q-integral on . From those applications, we derive some new interesting properties on the new family of Euler numbers and polynomials. That is, a systemic study of the class of Sheffer sequences in connection with generating function of the weighted q-Euler polynomials is given in the present paper. [Copyright &y& Elsevier]

To date, many interesting subclasses of analytic functions involving symmetrical points and other well celebrated domains have been investigated and studied. The aim of our present investigation is to make use of certain Janowski functions and a Mathieu-type series to define a new subclass of analytic (or invariant) functions. Our defined function class is symmetric under rotation. Some useful results like Fekete-Szegö functional, a number of sufficient conditions, radius problems, and results related to partial sums are derived. [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]

We consider weighted q-Genocchi numbers and polynomials. We investigated some interesting properties of the weighted q-Genocchi numbers related to weighted q-Bernstein polynomials by using fermionic p-adic integrals on ℤp. [ABSTRACT FROM AUTHOR]

In this paper, we test the dynamic symmetric and asymmetric causality relationship between the ecological footprint and trade openness in G7 countries by suggesting a new bootstrap panel causality test based on seemingly unrelated regressions. We analyzed the time-varying behavior of the symmetric and asymmetric panel causality relationship test to reveal the instabilities in the causality relationships. The obtained results suggest a relationship between the trade openness and ecological footprint in some years of the analysis period and between the negative and positive shocks. [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 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]

The logarithmic functions have been used in a verity of areas of mathematics and other sciences. As far as we know, no one has used the coefficients of logarithmic functions to determine the bounds for the third Hankel determinant. In our present investigation, we first study some well-known classes of starlike functions and then determine the third Hankel determinant bound for the logarithmic coefficients of certain subclasses of starlike functions that also involve the sine functions. We also obtain a number of coefficient estimates. Some of our results are shown to be sharp. [ABSTRACT FROM AUTHOR]

Keeping in view the latest trends toward quantum calculus, due to its various applications in physics and applied mathematics, we introduce a new subclass of meromorphic multivalent functions in Janowski domain with the help of the q-differential operator. Furthermore, we investigate some useful geometric and algebraic properties of these functions. We discuss sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikeness, radius of convexity, inclusion property, and convex combinations via some examples and, for some particular cases of the parameters defined, show the credibility of these results. [ABSTRACT FROM AUTHOR]

This paper deals with Al-Salam fractional q-integral operator and its application to certain q-analogues of Bessel functions and power series. Al-Salam fractional q-integral operator has been applied to various types of q-Bessel functions and some power series of special type. It has been obtained for basic q-generating series, q-exponential and q-trigonometric functions as well. Various results and corollaries are provided as an application to this theory. [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]

Several fractional calculus operators have been introduced and investigated. In this sequence, we aim to establish the Marichev-Saigo-Maeda (MSM) fractional calculus operators and Caputo-type MSM fractional differential operators of extended Mittag-Leffler function (EMLF). We also investigate the statistical distribution associated with the EMLF. Finally, we derive some of the particular cases of the main results. [ABSTRACT FROM AUTHOR]

We aim to introduce arbitrary complex order Hermite-Bernoulli polynomials and Hermite-Bernoulli numbers attached to a Dirichlet character χ and investigate certain symmetric identities involving the polynomials, by mainly using the theory of p-adic integral on Z p . The results presented here, being very general, are shown to reduce to yield symmetric identities for many relatively simple polynomials and numbers and some corresponding known symmetric identities. [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]