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Register a new userA uniqueness theorem of complex $q$-difference operator
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XiaoHuang Huang
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In this paper, we study the uniqueness of zero-order entire functions and their difference. We have proved: Let $f(z)$ be a nonconstant entire function of zero order, let $q eq0, eta$ be two finite complex numbers, and let $a$ and $b$ be two distinct complex numbers. If $f(z)$ and $Delta_{q,eta}f(z)$ share $a$, $b$ IM, then $fequiv Delta_{q,eta}f$.
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In this paper, we study the uniqueness of meromporphic functions and their difference operators. In particular, We have proved: Let $f$ be a nonconstant entire function on $mathbb{C}^{n}$, let $etain mathbb{C}^{n}$ be a nonzero complex number, and let $a$ and $b$ be two distinct complex numbers in $mathbb{C}^{n}$. If $$varlimsup_{rrightarrowinfty}frac{logT(r,f)}{r}=0,$$ and if $f(z)$ and $(Delta_{eta}^{n}f(z))^{(k)}$ share $a$ CM and share $b$ IM, then $f(z)equiv(Delta_{eta}^{n}f(z))^{(k)}$.
It is shown that if equation begin{equation*} f(z+1)^n=R(z,f), end{equation*} where $R(z,f)$ is rational in both arguments and $deg_f(R(z,f)) ot=n$, has a transcendental meromorphic solution, then the equation above reduces into one out of several types of difference equations where the rational term $R(z,f)$ takes particular forms. Solutions of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, exponential type functions or functions which are solutions to a certain autonomous first-order difference equation having meromorphic solutions with preassigned asymptotic behavior. These results complement our previous work on the case $deg_f(R(z,f))=n$ of the equation above and thus provide a complete difference analogue of Steinmetz generalization of Malmquists theorem. Finally, a description of how to simplify the classification in the case $deg_f(R(z,f))=n$ is given by using the new methods introduced in this paper.
We consider a family of solutions of $q-$difference Riccati equation, and prove the meromorphic solutions of $q-$difference Riccati equation and corresponding second order $q-$difference equation are concerning with $q-$gamma function. The growth and value distribution of differences on solutions of $q-$difference Riccati equation are also investigated.
This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions of zero order in the complex plane $mathbb{C}$. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem and five-value theorem in sense of Jackson $q$-difference operator. By using this theory, we investigate the growth of entire solutions of linear Jackson $q$-difference equations $D^{k}_{q}f(z)+A(z)f(z)=0$ with meromorphic coefficient $A,$ where $D^k_q$ is Jackson $k$-th order difference operator, and estimate the logarithmic order of some $q$-special functions.
We construct the Baxters operator and the corresponding Baxters equation for a quantum version of the Ablowitz Ladik model. The result is achieved by looking at the quantum analogue of the classical Backlund transformations. For comparison we find the same result by using the well-known Bethe ansatz technique. General results about integrable models governed by the same r-matrix algebra will be given. The Baxters equation comes out to be a q-difference equation involving both the trace and the quantum determinant of the monodromy matrix. The spectrality property of the classical Backlund transformations gives a trace formula representing the classical analogue of the Baxters equation. An explicit q-integral representation of the Baxters operator is discussed.
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