Solutions Of Q-difference Equations Research Articles

Singular Direction and q-Difference Operator of Meromorphic Functions

We study the common singular direction problem of meromorphic function for q-difference version operator; some criterions of the existence of common singular direction have been established. Further, the common singular direction of solutions of q-difference equations is also discussed in this paper.

A note on q-difference equations for Ramanujan’s integrals

This short paper derives the relationship between solutions of q-difference equations and generating functions for q-orthogonal polynomials. The key of the method is to obtain the expression of certain q-orthogonal polynomials as solutions of q-difference equations. In addition, we show how to generalize Ramanujan’s integrals by the technique of q-difference equation. More over, we find two generalized q-Chu–Vandermonde formulas from the perspective of the method of q-difference equations.

Q-difference equations for Askey–Wilson type integrals via q-polynomials

In this paper, we show how to deduce Askey–Wilson type integrals by the method of q-difference equation. In addition, we construct the relation between bilinear generating functions and solutions of q-difference equation. More over, we generalize Bailey's ψ66 summation from the perspective of q-integral by the method of q-difference equation. At last, we deduce U(n+1) type generating functions for Al-Salam–Carlitz polynomials by the method of q-difference equation.

A note on q-difference equations for Cigler’s polynomials

Viewing from polynomials solutions of q-difference equations makes certain generating functions calculated easily. In this short paper, we build the relations between Cigler’s polynomials and q-difference equations. In addition, we deduce generating functions and bilinear generating functions for Cigler’s polynomials. More over, we gain multilinear and mixed generating functions for Cigler’s polynomials. At last, we gain type generating functions for Cigler’s polynomials.

Legendre wavelet – quasilinearization technique for solving q-difference equations

Recently, various fixed point theorems have been used to prove the existence and uniqueness of the solutions for q-difference equations. In this paper, we obtain the existence and uniqueness theorems for a q-initial and a q-boundary value problem using the classical Newton’s method. Making use of the main theorems, a Legendre wavelet technique has been proposed to solve the q-difference equations numerically. The numerical simulation shows that the proposed scheme produces higher accuracy and is very straightforward to apply.

Two-circles theorem, q-periodic functions and entangled qubit states

For arbitrary hydrodynamic flow in circular annulus we introduce the two circle theorem, allowing us to construct the flow from a given one in infinite plane. Our construction is based on q-periodic analytic functions for complex potential, leading to fixed scale-invariant complex velocity, where q is determined by geometry of the region. Self-similar fractal structure of the flow with q-periodic modulation as solution of q-difference equation is studied. For one point vortex problem in circular annulus by fixing singular points we find solution in terms of q-elementary functions. Considering image points in complex plane as a phase space for qubit coherent states we construct Fibonacci and Lucas type entangled N-qubit states. Complex Fibonacci curve related to this construction shows reach set of geometric patterns.

ZERO ORDER ESTIMATES FOR ANALYTIC FUNCTIONS

In this article we develop an important tool in transcendental number theory. More precisely, we study multiplicity estimates (or multiplicity lemmas) for analytic functions. Our main theorem reduces multiplicity estimates at zero to the study of ideals in polynomial ring stable under an appropriate map. In particular, in the case of algebraic morphisms this result gives a new link between the theory of polarized algebraic dynamical systems and transcendental number theory. Specialized to the case of differential operators this theorem improves Nesterenko's conditional result on solutions of systems of differential equations. We also deduce an analog of Nesterenko's theorem for Mahler's functions and for solutions of q-difference equations. Further, analyzing stable ideals we prove the unconditional optimal result in the case of linear functional systems of generalized Mahler's type. The latter result generalizes a famous theorem of Nishioka (1986) previously conjectured by Mahler (1969). This new multiplicity estimate allows to prove new results on algebraic independence and on measures of algebraic independence, as done in Zorin (2010 and 2011).

On Identities Arising from Solutions of q-Difference Equations and Some Interpretations in Number Theory

On Identities Arising from Solutions of q-Difference Equations and Some Interpretations in Number Theory