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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 seco
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
We investigate arithmetic properties of values of the entire function $$ F(z)=F_q(z;lambda)=sum_{n=0}^inftyfrac{z^n}{prod_{j=1}^n(q^j-lambda)}, qquad |q|>1, quad lambda otin q^{mathbb Z_{>0}}, $$ that includes as special cases the Tschakaloff functio
We give a $q$-analog of middle convolution for linear $q$-difference equations with rational coefficients. In the differential case, middle convolution is defined by Katz, and he examined properties of middle convolution in detail. In this paper, we
We develop the theory of $p$-adic confluence of $q$-difference equations. The main result is the surprising fact that, in the $p$-adic framework, a function is solution of a differential equation if and only if it is solution of a $q$-difference equa