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In this paper, we introduce a new class of symmetric orthogonal polynomials, which generalizes the q-Dunkl-classical polynomials and also provides new characterizations of symmetric q-semiclassical orthogonal polynomials. Both the notion of a class and a criterion for determining the class will be given. As an application, several examples of q-Dunkl-semiclassical forms of the class $ s \leqq 2 $ s ≦ 2 are highlighted. [ABSTRACT FROM AUTHOR]

Two new subclasses of the class of bi-Bazilevič functions, which are related to the Fibonacci-number series and the square-root functions, are introduced and studied in this article. Under a special choice of the parameter involved, these two classes of Bazilevič functions reduce to two new subclasses of star-like biunivalent functions related with the Fibonacci-number series and the square-root functions. Using the Faber polynomial expansion (FPE) technique, we find the general coefficient bounds for the functions belonging to each of these classes. We also find bounds for the initial coefficients for bi-Bazilevič functions and demonstrate how unexpectedly these initial coefficients behave in relation to the square-root functions and the Fibonacci-number series. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce and study the concepts of statistical convergence and statistical summability for martingale sequences of random variables via deferred Cesaro ` mean. We then establish an inclusion theorem concerning the relation between these two beautiful and potentially useful concepts. Also, based upon our proposed notions, we state and prove new Korovkin-type approximation theorems with algebraic test functions for a martingale sequence over a Banach space. Moreover, we demonstrate that our theorems effectively extend and improve most (if not all) of the previously existing results (in statistical and classical versions). Finally, by using the generalized Bernstein polynomials, we present an illustrative example of a martingale sequence in order to demonstrate that our established theorems are stronger than their traditional and statistical versions. [ABSTRACT FROM AUTHOR]

The main purpose of this article is to investigate a novel set of (orthogonal) basis functions for treating a class of multi-order fractional pantograph differential equations (MOFPDEs) computationally. These polynomials, denoted by Sn(x) and called special polynomials, were first discovered in a study of a certain family of isotropic turbulence fields. They are expressible in terms of the generalized Laguerre polynomials and are related to the Bessel and Srivastava–Singhal polynomials. Unlike the Laguerre polynomials, all coefficients of the special polynomials are positive. We further introduce the fractional order of the special polynomials and use them along with some suitable collocation points in a special matrix technique to treat fractional-order MOFPDEs. Moreover, the convergence analysis of these polynomials is established. Through five example applications, the utility and efficiency of the present matrix approach are demonstrated and comparisons with some existing numerical schemes have been performed in this class. [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]

The purpose of this article is to introduce a new subclass HΣ(δ, λ, m, θ, x) of analytic and bi-univalent functions by using the Horadam polynomials, which is associated with the Pascal distribution series and to investigate the bounds for |a2| and |a3|, where a2, a3 are the initial Taylor-Maclaurin coefficients. Further we obtain the Fekete-Szeg¨o inequality for functions in the class HΣ(δ, λ, m, θ, x) which we have introduced here. [ABSTRACT FROM AUTHOR]

One of the tools and techniques concerned with the stability of nonlinear waves is the Evans function which is an analytic function whose zeros give the eigenvalues of the linearized operator. Here, in this paper, we propose a direct approach, which is based essentially upon constructing the eigenfunction solution of the perturbed equation based upon the topological invariance in conjunction with usage of the Legendre polynomials, which have presumably not considered in the literature thus far. The associated Legendre eigenvalue problem arising from the stability analysis of traveling waves solutions is systematically studied here. The present work is of considerable interest in the engineering sciences as well as the mathematical and physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product. This can be controlled by depicting the initial concentration of the reactant, which is determined by its value at the bifurcation point. This analysis leads to the point separating stable and unstable solutions. As far as chemical reactions are described by reaction-diffusion equations, this specific concentration can be found mathematically. On the other hand, the study of stability analysis of solutions may depict whether or not a soliton pulse is well-propagated in fiber optics. This can, and should, be carried out by finding the solutions of the coupled nonlinear Schrödinger equations and by analyzing the stability of these solutions. [ABSTRACT FROM AUTHOR]

In this article, the authors establish the recurrence relations and q-difference equations for the 2-iterated q-Appell polynomials. The recurrence relations and the q-difference equations for the 2-iterated q-Bernoulli polynomials, the q-Euler polynomials and the q-Genocchi polynomials are also derived. An analogous study of certain mixed type q-special polynomials is also presented. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce new families of the q-Fibonacci and q-Lucas polynomials, which are represented here as the incomplete q-Fibonacci polynomials Fnk (x, s, q) and the incomplete q-Lucas polynomials Lnk (x, s, q), respectively. These polynomials provide the q-analogues of the incomplete Fibonacci and Lucas numbers. We give several properties and generating functions of each of these families q-polynomials. We also point out the fact that the results for the q-analogues which we consider in this article for 0 < q < 1 can easily be translated into the corresponding results for the (p, q)-analogues (with 0 < q < p ≥ 1) by applying some obvious parametric variations, the additional parameter p being redundant. [ABSTRACT FROM AUTHOR]

Motivated by the demonstrated potential for their applications in various research areas such as those in mathematical, physical, engineering, and statistical sciences, our main object in this paper is to introduce and investigate a fractional integral operator that contains a certain generalized multi‐index Mittag‐Leffler function in its kernel. In particular, we establish some interesting expressions for the composition of such well‐known fractional integral and fractional derivative operators as (for example) the Riemann‐Liouville fractional integral and fractional derivative operators, the Hilfer fractional derivative operator, and the above‐mentioned fractional integral operator with the generalized multi‐index Mittag‐Leffler function in its kernel. The main findings in this paper are shown to generalize the results that were derived earlier by Kilbas et al and Srivastava et al. Finally, in this paper, we derive integral representations for the product of 2 generalized multi‐index Mittag‐Leffler functions in terms of the familiar Fox‐Wright hypergeometric function. [ABSTRACT FROM AUTHOR]

Motivated by the fact that fractional q-integrals play important roles in numerous areas of mathematical, physical and engineering sciences, it is natural to consider the corresponding iterated fractional q-integrals. The main object of this paper is to define these iterated fractional q-integrals, to build the relations between iterated fractional q-integrals and certain families of generating functions for q-polynomials and to generalize two fractional q-identities which are given in a recent work [Fract. Calc. Appl. Anal. 10 (2007), 359–373]. As applications of the main results presented here, we deduce several bilinear generating functions, Srivastava-Agarwal type generating functions, multilinear generating functions and U(n + 1) type generating functions for the Rajković-Marinković-Stanković polynomials. [ABSTRACT FROM AUTHOR]

Recently, the notion of positive linear operators by means of basic (or q-) Lagrange polynomials and 𝒜 {\mathcal{A}} -statistical convergence was introduced and studied in [M. Mursaleen, A. Khan, H. M. Srivastava and K. S. Nisar, Operators constructed by means of q-Lagrange polynomials and A-statistical approximation, Appl. Math. Comput. 219 2013, 12, 6911–6918]. In our present investigation, we introduce a certain deferred weighted 𝒜 {\mathcal{A}} -statistical convergence in order to establish some Korovkin-type approximation theorems associated with the functions 1, t and t 2 {t^{2}} defined on a Banach space C ⁢ [ 0 , 1 ] {C[0,1]} for a sequence of (presumably new) positive linear operators based upon ( p , q ) {(p,q)} -Lagrange polynomials. Furthermore, we investigate the deferred weighted 𝒜 {\mathcal{A}} -statistical rates for the same set of functions with the help of the modulus of continuity and the elements of the Lipschitz class. We also consider a number of interesting special cases and illustrative examples in support of our definitions and of the results which are presented in this paper. [ABSTRACT FROM AUTHOR]

Using the Tremblay fractional derivative operator in the complex domain, we introduce and investigate a new class of analytic and bi-univalent functions in the open unit disk. We use the Faber polynomial expansions to obtain upper bounds for the general coefficients of such functions subject to a gap series condition as well as obtaining bounds for their first two coefficients. [ABSTRACT FROM AUTHOR]

In their recent investigation involving differential operators for the generalized Lagrange polynomials, Chan et. al. [3] encountered and proved a certain summation identity and several other results for the Lagrange polynomials in several variables, which are popularly known in the literature as the Chan-Chyan-Srivastava polynomials. These multivariable polynomials have been studied systematically and extensively in the literature ever since then (see, for example, [1], [4], [9], [11], [12] and [13]). In the present paper, we investigate umbral calculus presentations of the Chan-Chyan-Srivastava polynomials and also of their substantially more general form, the Erkuș-Srivastava polynomials [9]. Some other closely-related results are also considered. [ABSTRACT FROM AUTHOR]

In some recent investigations involving differential operators for a general family of Lagrange polynomials, Chenet al.[Some new results for the Lagrange polynomials in several variables, ANZIAM J. 49 (2007), pp. 243–258] encountered and proved a certain summation identity for the Chan–Chyan–Srivastava polynomials. With a view to extending and generalizing the aforementioned summation identity, we derive the corresponding results for one class of hypergeometric polynomials, the Jacobi polynomials, the extended Jacobi polynomials, the Laguerre polynomials, the Hermite polynomials, the Lagrange–Hermite polynomials and the Erkuş–Srivastava polynomials. [ABSTRACT FROM PUBLISHER]

The main purpose of this paper is to construct generating functions for some families of Genocchi type polynomials. By applying the derivative operator to these generating functions, we define a function which interpolates these polynomials at negative integers. Among other results, we prove a multiplication theorem for these polynomials. Furthermore, we give several other identities and applications associated with these and related polynomials and their interpolation functions. [ABSTRACT FROM PUBLISHER]

The main object of this paper is to consider the asymptotic distribution of the zeros of certain classes of the Gauss hypergeometric polynomials. Some classical analytic methods and techniques are used here to analyze the behavior of the zeros of the Gauss hypergeometric polynomials, 2F1(-n, a;-n + b; z), where n is a nonnegative integer. Owing to the connection between the classical Jacobi polynomials and the Gauss hypergeometric polynomials, we prove a special case of a conjecture made by Martínez-Finkelshtein, Martínez-González and Orive. Numerical evidence and graphical illustrations of the clustering of the zeros on certain curves are generated by Mathematica (Version 4.0). [ABSTRACT FROM AUTHOR]

The main purpose of this article is to provide probabilistic proofs of the relationships between the generalized Bernoulli (or Nörlund) polynomials B(α)n (x) and the generalized Euler polynomials E(α)n (x) of (real or complex) order α and degree n in x, which were proved recently by Srivastava and Pintér [11]. Some other approaches to these relationships and their seemingly interesting generalizations are also investigated. [ABSTRACT FROM AUTHOR]

The main object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. Each of the integral representations, which are derived in this paper, may be viewed also as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials. [ABSTRACT FROM AUTHOR]

The present investigation is a continuation of the works initiated by Srivastava [A contour integral involving Fox's H-function, Indian J. Math. 14 (1972), pp. 1-6] and followed by several papers such as (among others) [A. Altın, E. Erkus, and M.A. Ozarslan, Families of linear generating functions for polynomials in two variables, Integral Transforms Spec. Funct. 17 (2006), pp. 315-320; B. Gonzalez, J. Matera, and H.M. Srivastava, Some q-generating functions and associated generalized hypergeometric polynomials, Math. Comput. Modelling 34 (2001), pp. 133-175; S.-D. Lin, Y.-S. Chao, and H.M. Srivastava, Some families of hypergeometric polynomials and associated integral representations, J. Math. Anal. Appl. 294 (2004), pp. 399-411; S.-D. Lin, H.M. Srivastava, and P.-Y. Wang, Some families of hypergeometric transformations and generating relations, Math. Comput. Modelling 36 (2002), pp. 445-459; E. Ozergin, M.A. Ozarslan, and H.M. Srivastava, Some families of generating functions for a class of bivariate polynomials, Math. Comput. Modelling 50 (2009), pp. 1113-1120]. In this study, we introduce a family of three-variable polynomials and derive a number of two-sided linear generating functions between these polynomials and another family of two-variable polynomials. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials. Several other results of the above-mentioned types are also considered. [ABSTRACT FROM AUTHOR]

Let [image omitted] be an infinite set of linear arbitrary functionals. Also let [image omitted] be a certain set of basis functions such that [image omitted] δi, j being (as usual) the Kronecker delta. In this paper, by defining the sequence of basis functions: [image omitted] in which u(x) is an arbitrary real function, [image omitted] we construct a new functional expansion for analytic functions and then introduce three important subclasses of this functional expansion that correspond, respectively, to the orthogonal polynomial expansion, the Taylor series expansion, and the Fourier trigonometric expansion. [ABSTRACT FROM AUTHOR]

By introducing the Taylor polynomials, a significantly refined version of Wilker's inequality is established. The result is then used to obtain several substantially more refined inequalities of the Wilker type. [ABSTRACT FROM AUTHOR]

In this paper, the classical Jordan's inequality is studied once again, a new sharpened and generalized version of Jordan's inequality is given by means of polynomial representation, and the result thus derived is then used to obtain several substantially more refined inequalities of Jordan type. Finally, an application of the results presented in this paper toward the improvement of the Yang Le inequality is considered. [ABSTRACT FROM AUTHOR]

Finite sums of powers of cosecants appear in a wide range of physical problems. We, through a unified approach which uses contour integrals and residues, establish the summation formulas for two general families of such sums. One of them is the family which was first studied and summed in closed form by Dowker [Phys. Rev. D 36, 3095 (1987)], while the other is related to it and has not been studied before. Our summation formulas of the Dowker sums involve only the Stirling numbers of the first kind and the (ordinary) Bernoulli polynomials and numbers, unlike the earlier summation formulas in which either the higher-order Bernoulli numbers and polynomials or the multiple sums involving the Bernoulli numbers and their products, were used. A great deal of other (known or presumably new) closed-form summations follows as straightforward corollaries to these formulas. Among them are two special cases of the celebrated Verlinde’s formula and numerous sums encountered in various physical problems by McCoy and Orrick [J. Stat. Phys. 83, 839 (1996)], Gervois and Mehta [J. Math. Phys. 36, 5098 (1995)], and Henkel and Lacki [Phys. Lett. A 138, 105 (1989)]. [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]