Partial sums of generalized harmonic starlike univalent functions generated by a (p,q)−Ruscheweyh-type harmonic differential operator

Let H denote the class of complex-valued harmonic functions f defined in the open unit disc D and normalized by f(0)=fz(0)-1=0.In this paper,we define a new generalized subclass of H associated with the(p,q)-Ruscheweyh-type harmonic differential operator in D.We first obtain a sufficient coefficient condition that guarantees that a function f in H is sense-preserving harmonic univalent in D and belongs to the aforementioned class.Using this coefficient condition,we then examine ratios of partial sums of f in H.In all cases the results are sharp.In addition,the results so obtained generalize the related works of some authors,and many other new results are obtained.

Toeplitz and Translation Operators on the q-Fock Spaces

In this work, we introduce a class of Hilbert spaces Fq of entire functions on the disk , , with reproducing kernel given by the q-exponential function eq(z);and we prove some properties concerning Toeplitz operators on this space. The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Dq by and the q-Derivative operator on the Fock space Fq;and we prove that these operators are adjoint-operators and continuous from this space into itself. Lastly, we study a generalized translation operators and a Weyl commutation relations on Fq .

STANCU POLYNOMIALS BASED ON THE Q-INTEGERS

A new generalization of Stancu polynomials based on the q-integers and a nonnegative integer s is firstly introduced in this paper. Moreover, the shape-preserving and convergence properties of these polynomials are also investigated.

On some properties of the bibasic Humbert hypergeometric functions Ξ_(1) and Ξ_(2)

The main object of this paper is to deduce the bibasic Humbert functions Ξ_(1) and Ξ_(2)Some interesting results and elementary summations technique that was successfully employed,q-recursion,q-derivatives relations,the q-differential recursion relations,the q-integral representations for Ξ_(1) and Ξ_(2)are given.The summation formula derives a list of p-analogues of transformation formulas for bibasic Humbert functions that have been studied,also some hypergeometric functions properties of some new interesting special cases have been given.

Numerical Approximation of Quantum-Integrals Using the Appropriate Nodes and Weights

In this paper, we present a procedure for the numerical q-calculation of the q-integrals based on appropriate nodes and weights which are determined such that the error of q-integration is mini-mized;a system of linear and nonlinear set of equations respectively are prepared to obtain the nodes and weights simultaneously;the error of q-integration is considered to be minimized under this condition;finally some application and numerical examples are given for comparison with the exact solution. At the end, the related tables of approximations are presented.

A Generalization of the Concept of q-fractional Integrals

In this paper, we consider the fractional q-integral with variable lower limit of integration. We prove the semigroup property of these integrals, and a formula of Leibniz type. Finally, we evaluate fractional q-integrals of some functions. The consideration of q-exponential function in that sense leads to q-analogs of Mittag-Leffier function.

Symmetric q-Deformed KP Hierarchy

Based on the analytic property of the symmetric q-exponent e_q(x),a new symmetric q-deformed Kadomtsev-Petviashvili(q-KP for short) hierarchy associated with the symmetric q-derivative operator α_q is constructed.Furthermore,the symmetric q-CKP hierarchy and symmetric q-BKP hierarchy are defined.The authors also investigate the additional symmetries of the symmetric q-KP hierarchy.

A q-operational equation and the Rogers-Szegő polynomials

By solving a q-operational equation with formal power series,we prove a new q-exponential operational identity.This operational identity reveals an essential feature of the Rogers-Szegő polynomials and enables us to develop a systematic method to prove the identities involving the Rogers-Szegő polynomials.With this operational identity,we can easily derive,among others,the q-Mehler formula,the q-Burchnall formula,the q-Nielsen formula,the q-Doetsch formula,the q-Weisner formula,and the Carlitz formula for the Rogers-Szegő polynomials.This operational identity also provides a new viewpoint on some other basic q-formulas.It allows us to give new proofs of the q-Gauss summation and the second and third transformation formulas of Heine and give an extension of the q-Gauss summation.

Approximation by Complex q-Durrmeyer Polynomials in Compact Disks

In this paper, the order of approximation and Voronovskaja type results with quantitative estimate for complex q-Durrmeyer polynomials attached to analytic functions on compact disks are obtained.

On the existence and uniqueness of a nonlinear q-difference boundary value problem of fractional order

In this research collection,we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional q-difference equation having the given form c D^(ζ)_(q) v(t)−h(t,v(t))=0,0≤t≤1,α_(1)v(0)+β_(1)D_(q)v(0)=v(η1),α_(2)v(1)−β_(2)D_(q)v(1)=v(η2),where 1<ζ≤2,(η1,η2)∈(0,1)^(2),α_(i),β_(i)∈R(i=1,2),h∈C([0,1]×R,R)and c Dζq represents the Caputo-type nonclassical q-derivative of orderζ.We use well-known principal of Banach contraction,and Leray–Schauder nonlinear alternative to vindicate the unique solution existence of the given problem.Regarding the applications,some examples are solved to justify our outcomes.