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Register a new user$q-$difference equations concerning with $q-$gamma function
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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.
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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.
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 $fequiv Delta_{q,eta}f$.
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 function ($lambda=0$) and the $q$-exponential function ($lambda=1$). In particular, we prove the non-quadraticity of the numbers $F_q(alpha;lambda)$ for integral $q$, rational $lambda$ and $alpha otin-lambda q^{mathbb Z_{>0}}$, $alpha e0$.
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 define a $q$-analog of middle convolution. Moreover, we show that it also can be expressed as a $q$-analog of Euler transformation. The $q$-middle convolution transforms Fuchsian type equation to Fuchsian type equation and preserves rigidity index of $q$-difference equations.
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 equation. This fact implies an equivalence, called ``Confluence, between the category of differential equations and those of $q$-difference equations. We obtain this result by introducing a category of ``sheaves on the disk $mathrm{D}^-(1,1)$, whose stalk at 1 is a differential equation, the stalk at $q$ is a $q$-difference equation if $q$ is not a root of unity $xi$, and the stalk at a root of unity is a mixed object, formed by a differential equation and an action of $sigma_xi$.
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