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.

Note on a New Construction of Kantorovich Form q-Bernstein Operators Related to Shape Parameter λ

The main purpose of this paper is to introduce some approximation properties of a Kantorovich kind q-Bernstein operators related to B′ezier basis functions with shape parameterλ∈[−1,1].Firstly,we compute some basic results such as moments and central moments,and derive the Korovkin type approximation theorem for these operators.Next,we estimate the order of convergence in terms of the usual modulus of continuity,for the functions belong to Lipschitz-type class and Peetre’s K-functional,respectively.Lastly,with the aid of Maple software,we present the comparison of the convergence of these newly defined operators to the certain function with some graphical illustrations and error estimation table.

On <i>q</i>-Deformed Calculus in Quantum Geometry

The relation between noncommutative (or quantum) geometry and themathematics of spacesis in many ways similar to the relation between quantum physicsand classical physics. One moves from the commutative algebra of functions on a space (or a commutative algebra of classical observable in classical physics) to a noncommutative algebra representing a noncommutative space (or a noncommutative algebra of quantum observables in quantum physics). The object of this paper is to study the basic rules governing q-calculus as compared with the classical Newton-Leibnitz calculus.

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.

(p,q)-Analogue of Mittag-Leffler Function with(p,q)-Laplace Transform

The aim of this paper is to define(p,q)-analogue of Mittag-Leffler Function,by using(p,q)-Gamma function.Some transformation formulae are also derived by using the(p,q)-derivative.The(p,q)-analogue for this function provides elegant generalization of q-analogue of Mittag-Leffler function in connection with q-calculus.Moreover,the(p,q)-Laplace Transform of the Mittag-Leffler function has been obtained.Some special cases have also been discussed.

An extensive analytical and numerical study of the generalized q-deformed Sinh-Gordon equation

In this paper,the q-deformed Sinh-Gordon equation is solved analytically using a new general form based on the extended tanh approach.The numerical solutions of the equation is obtained using a b-spline finite element method.Also,we present numerous figures to demonstrate the various solitons propagation patterns.This type of equation has not been previously dealt with in such ways,whether analytical or numerical.This study is very useful in studying several physical systems that have lost their symmetry.