Mathematics Subject Classification: 33D60, 33E12, 26A33Based on the fractional q–integral with the parametric lower limit of integration, we consider the fractional q–derivative of Caputo type. Especially, its applications to q-exponential functions allow us to introduce q–analogues of the Mittag–Leffler function. Vice versa, those functions can be used for defining generalized operators in fractional q–calculus.

This work is devoted to the study of a Cauchy problem for a certain family of q-difference-differential equations having Fuchsian and irregular singularities. For given formal initial conditions, we first prove the existence of a unique formal power series X̂(t,z) solving the problem. Under appropriate conditions, q-Borel and q-Laplace techniques (firstly developed by J.-P. Ramis and C. Zhang) help us in order to construct actual holomorphic solutions of the Cauchy problem whose q-asymptotic expansion...

Let $M=G/H$ be a real symmetric space and $𝔤=𝔥+𝔮$ the corresponding decomposition of the Lie algebra. To each open $H$-invariant domain $D_{𝔮}\subseteq i𝔮$ consisting of real ad-diagonalizable elements, we associate a complex manifold $\Xi \left(D_{𝔮}\right)$ which is a curved analog of a tube domain with base $D_{𝔮}$, and we have a natural action of $G$ by holomorphic mappings. We show that $\Xi \left(D_{𝔮}\right)$ is a Stein manifold if and only if $D_{𝔮}$ is convex, that the envelope of holomorphy is schlicht and that $G$-invariant plurisubharmonic functions correspond to convex $H$-invariant...

The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation $$f^n\left(z\right)+P_d(z,f)=p_1\left(z\right){\mathrm{e}}^{{\alpha }_1\left(z\right)}+p_2\left(z\right){\mathrm{e}}^{{\alpha }_2\left(z\right)},$$ where $P_d(z,f)$ is a difference-differential polynomial in $f\left(z\right)$ of degree $d\le n-1$ with small functions of $f\left(z\right)$ as its coefficients, $p_1$, $p_2$ are nonzero rational functions and ${\alpha }_1$, ${\alpha }_2$ are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.