Related papers: $q-$difference equations concerning with $q-$gamma…
In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for…
In this paper, we have found that some certain Fermat-type shift and difference equations have the meromorphic solutions generated by Riccati type functions. Also we have solved the open problems posed by Liu and Yang (A note on meromorphic…
Earlier work introduced a method for obtaining indefinite $q$-integrals of $q$-special functions from the second-order linear $q$-difference equations that define them. In this paper, we reformulate the method in terms of $q$-Riccati…
A q-difference analogue of the Painlev\'e III equation is considered. Its derivations, affine Weyl group symmetry, and two kinds of special function type solutions are discussed.
We consider the first order $q$-difference equation \begin{equation}\tag{\dag} f(qz)^n=R(z,f), \end{equation} where $q\not=0,1$ is a constant and $R(z,f)$ is rational in both arguments. When $|q|\not=1$, we show that, if $(\dag)$ has a zero…
Using the q-version of the Darboux transform we obtain the general solution of q-difference Riccati equation from a special one by the action of one-parameter group. This allows us to construct the solutions for the latge class of…
In this short paper, we show how to deduce several types of generating functions from Srivastava {\it et al} [Appl. Set-Valued Anal. Optim. {\bf 1} (2019), pp. 187-201.] by the method of $q$-difference equations. Moreover, we build…
In this paper, we continue to study the sharing value problems for higher order derivatives of meromorphic functions with its linear difference and $q$-difference operators. Some of our results generalize and improve the results of…
In the present paper, a general theory for the second-order matrix difference equation of bilateral type is discussed. We introduced the matrix $q$-Kummer equation of bilateral type and presented the $q$-Kummer matrix function as a series…
In this paper, the authors establish some inequalities involving the $q$-extension of the classical Gamma function. These inequalities provide bounds for certain ratios of the $q$-extended Gamma function. The procedure makes use of…
In this paper, we consider the behaviour, when $q$ goes to $1$, of the set of a convenient basis of meromorphic solutions of a family of linear $q$-difference equations. In particular, we show that, under convenient assumptions, such basis…
The paper discusses the summability of formal solutions of some linear q-difference-differential equations, and improves the previous result in [Tahara-Yamazawa, Opsucula Math. 35 (2015), 713-738].
From two q-summation formulas we deduce certain series expansion formulas involving the q-gamma function. With these formulas we can give q-analogues of series expansions for certain constants.
We discuss a method of constructing solution of the initial value problem for duffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered.
We introduce new generalizations of the Gamma and the Beta functions. Their properties are investigated and known results are obtained as particular cases.
Fractional $q$-extensions of some classical $q$-orthogonal polynomials are introduced and some of the main properties of the new defined functions are given. Next, a fractional $q$-difference equation of Gauss type is introduced and solved…
In this paper we study an analogue of the classical Riemann-Hilbert problem stated for the classes of difference and $q$-difference systems. The Birkhoff's existence theorem was generalized in this paper.
In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the…
Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit…
Riccati's differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati's differential equation with the Cole-Hopf transform shows a relation between the first order evolution…