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 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.展开更多
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 ...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 .展开更多
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 al...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.展开更多
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 linea...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.展开更多
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 d...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.展开更多
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-integ...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.展开更多
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.
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_...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.展开更多
基金Supported by the National Natural Science Foundation of China(11601266)the Natural Science Foundation of Fujian Province of China(2020J01783)。
文摘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.
文摘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 .
文摘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.
文摘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.
基金supported by National Natural Science Foundation of China (Grant No.11971173)Science and Technology Commission of Shanghai Municipality (Grant No.13dz2260400)。
文摘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.
基金Supported by Ministry of Science,Technology and Development of Republic Serbia (Grant Nos.144023 and 144013)
文摘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.
文摘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.
文摘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.
