Q-difference Operators Research Articles

A New Family of Starlike Functions in a Circular Domain Involving a q-differential Operator

The main purpose of the present paper is to investigate a number of useful properties such as sufficiency criteria, distortion bounds, coefficient estimates, radius of starlikness and radius of convexity for a new subclass of analytic functions, which are defined here by means of a newly defined q-linear differential operator.

Diagonalization of the Heun-Askey-Wilson operator, Leonard pairs and the algebraic Bethe ansatz

An operator of Heun-Askey-Wilson type is diagonalized within the framework of the algebraic Bethe ansatz using the theory of Leonard pairs. For different specializations and the generic case, the corresponding eigenstates are constructed in the form of Bethe states, whose Bethe roots satisfy Bethe ansatz equations of homogeneous or inhomogenous type. For each set of Bethe equations, an alternative presentation is given in terms of ‘symmetrized’ Bethe roots. Also, two families of on-shell Bethe states are shown to generate two explicit bases on which a Leonard pair acts in a tridiagonal fashion. In a second part, the (in)homogeneous Baxter T-Q relations are derived. Certain realizations of the Heun-Askey-Wilson operator as second q-difference operators are introduced. Acting on the Q-polynomials, they produce the T-Q relations. For a special case, the Q-polynomial is identified with the Askey-Wilson polynomial, which allows one to obtain the solution of the associated Bethe ansatz equations. The analysis presented can be viewed as a toy model for studying integrable models generated from the Askey-Wilson algebra and its generalizations. As examples, the q-analog of the quantum Euler top and various types of three-sites Heisenberg spin chains in a magnetic field with inhomogeneous couplings, three-body and boundary interactions are solved. Numerical examples are given. The results also apply to the time-band limiting problem in signal processing.

Generalized distribution for analytic function classes associated with error functions and Bell numbers

The primary motivation of the paper is to study the generalized distribution for analytic function classes associated with error functions and Bell numbers. Further, the authors estimate the bounds for probability distribution series by means of q-difference operator using subordination principle.

Solving q-Virasoro constraints

We show how q-Virasoro constraints can be derived for a large class of (q, t)-deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral, in complete analogy with full-derivative insertions for beta -ensembles. From free field point of view, the models considered have zero momentum of the highest weight, which leads to an extra constraint T_{-1} mathcal {Z} = 0. We then show how to solve these q-Virasoro constraints recursively and comment on the possible applications for gauge theories, for instance calculation of (supersymmetric) Wilson loop averages in gauge theories on D^2 times S^1 and S^3.

An infrared bootstrap of the Schur index with surface defects

The infrared formula relates the Schur index of a 4d mathcal{N} = 2 theory to its wall-crossing invariant, a.k.a. BPS monodromy. A further extension of this formula, proposed by Córdova, Gaiotto and Shao, includes contributions by various types of line and surface defects. We study BPS monodromies in the presence of vortex surface defects of arbitrary vorticity for general class mathcal{S} theories of type A1 engineered by UV curves with at least one regular puncture. The trace of these defect BPS monodromies is shown to coincide with the action of certain q-difference operators acting on the trace of the (pure) 4d BPS monodromy. We use these operators to develop a “bootstrap” (of traces) of BPS monodromies, relying only on their infrared properties, thereby reproducing the very general ultraviolet characterization of the Schur index.

Bounds of the initial coefficient for sakaguchi function in the conical domain

In this paper, we consider a new class of sakaguchi type functions which is defined by Ruscheweyh q-differential operator. We investigated of co-efficient inequalities and other interesting properties of this class.

Inequalities for the Taylor Coefficients of Spiralike Functions Involving q-differential Operator

Making use of q-analogue of the well-known differential operator, we provide a formal extension of a bi-univalent spiralike and bi-univalent strongly spiralike functions. We obtain the inequalities for the Maclaurin-Taylor coefficients of the functions belonging to the defined subclasses. Further we have provided some applications of our main results.

A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator

Abstract Making use of Ruscheweyh q-differential operator, we define a new subclass of uniformly convex functions and corresponding subclass of starlike functions with negative coefficients. The main object of this paper is to obtain, coefficient estimates, closure theorems and extreme point for the functions belonging to this new class. The results are generalized to families with fixed finitely many coefficients.

The Heun–Askey–Wilson Algebra and the Heun Operator of Askey–Wilson Type

The Heun–Askey–Wilson algebra is introduced through generators $$\{{\textsf {X}},{\textsf {W}}\}$$ and relations. These relations can be understood as an extension of the usual Askey–Wilson ones. A central element is given, and a canonical form of the Heun–Askey–Wilson algebra is presented. A homomorphism from the Heun–Askey–Wilson algebra to the Askey–Wilson one is identified. On the vector space of the polynomials in the variable $$x=z+z^{-1}$$ , the Heun operator of Askey–Wilson type realizing $${\textsf {W}}$$ can be characterized as the most general second-order q-difference operator in the variable z that maps polynomials of degree n in $$x=z+z^{-1}$$ into polynomials of degree $$n+1$$ .

Estimation of initial Maclaurin coefficients of certain subclasses of bounded bi-univalent functions

In this paper, two bounded bi-univalent function subclasses were defined by using Salagean q-differential operator. The functions are defined in the open unit disc of complex plane. The main purpose is to determine some estimations on the initial Maclaurin coefficients for functions in these subclasses. Finally, the Fekete-Szegö inequalities for these are also obtained.

Fekete–Szegö inequalities for subclass of bi-univalent functions associated with Sălăgean type q-difference operator

In this paper, we introduce and investigate a new subclass of the function class $$\Sigma $$ of bi-univalent functions defined in the open unit disk, which are associated with the Sălăgean type q-difference operator and satisfy some subordination conditions. Furthermore, we find estimates on the Taylor–Maclaurin coefficients $$|a_{2}|$$ and $$|a_{3}|$$ for functions in the new subclass introduced here. Several (known or new) consequences of the results are also pointed out. Further we obtain Fekete–Szego inequality for the new function class.

A New Subclass of Analytic Functions Defined by Using Salagean q-Differential Operator

In our present investigation, we use the technique of convolution and quantum calculus to study the Salagean q-differential operator. By using this operator and the concept of the Janowski function, we define certain new classes of analytic functions. Some properties of these classes are discussed, and numerous sharp results such as coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, and integral mean inequalities of functions belonging to these classes are obtained and studied.

The q-Heun operator of big q-Jacobi type and the q-Heun algebra

The q-Heun operator of the big q-Jacobi type on the exponential grid is defined. This operator is the most general second-order q-difference operator that maps polynomials of degree n to polynomials of degree $$n+1$$. It is tridiagonal in bases made out of either q-Pochhammer or big q-Jacobi polynomials and is bilinear in the operators of the q-Hahn algebra. The extension of this algebra that includes the q-Heun operator as generator is described. Biorthogonal Pastro polynomials are shown to satisfy a generalized eigenvalue problem or equivalently to be in the kernel of a special linear pencil made out of two q-Heun operators. The special case of the q-Heun operator associated to the little q-Jacobi polynomials is also treated.

On Analytic Functions Involving the q-Ruscheweyeh Derivative

In this paper, we use concepts of q-calculus to introduce a certain type of q-difference operator, and using it define some subclasses of analytic functions. Inclusion relations, coefficient result, and some other interesting properties of these classes are studied.

Certain q-difference operators and their applications to the subclass of meromorphic q-starlike functions

The main aim of this work is to find some coefficient inequalities and sufficient condition for some subclasses of meromorphic starlike functions by using q-difference operator. Here we also define the extended Ruscheweyh differential operator for meromorphic functions by using q-difference operator. Several properties such as coefficient inequalities and Fekete-Szego functional of a family of functions are investigated.

Generalized q-Laguerre type polynomials and q-partial differential equations

In this paper we define the q-Laguerre type polynomials Un(x; y; z; q), which include q-Laguerre polynomials, generalized Stieltjes-Wigert polynomials, little q-Laguerre polynomials and q-Hermite polynomials as special cases. We also establish a generalized q-differential operator, with which we build the relations between analytic functions and Un(x; y; z; q) by using certain q-partial differential equations. Therefore, the corresponding conclusions about q-Laguerre polynomials, little q-Laguerre polynomials and q-Hermite polynomials are gained as corollaries. As applications, some generating functions and generalized Andrews-Askey integral formulas are given in the final section.

Some homogeneous q-difference operators and the associated generalized Hahn polynomials

In this paper, we first construct the homogeneous $q$-shift operator $\widetilde{E}(a,b;D_{q})$ and the homogeneous $q$-difference operator $\widetilde{L}(a,b; \theta_{xy})$. We then apply these operators in order to represent and investigate generalized Cauchy and a general form of Hahn polynomials. We derive some $q$-identities such as: generating functions, extended generating functions, Mehler's formula and Roger's formula for these $q$-polynomials.

Finite Series Solution Arising from Three-Dimensional q-Difference Equation

In this article, we introduce a three-dimensional q-difference operator with its equation. Also we derive certain theorems using three-dimensional q-difference operator. Suitable examples veri ed by MATLAB are inserted to illustrate our findings.

Bi-Bazilevič functions of complex order involving Ruscheweyh type q-difference operator

Abstract In this paper, we define a new subclass of bi-univalent functions involving q-difference operator in the open unit disk. For functions belonging to this class, we obtain estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|.

A note on q-difference operator and related limit direction

The growth of entire functions under the q-difference operators is studied in this paper, and then some properties of Julia set of entire functions under the higher order q-difference operators are obtained.