### Anharmonic oscillators with infinitely many real eigenvalues and $\mathrm{\mathcal{P}\mathcal{T}}$-symmetry.

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This is an expository article, based on the talk with the same title, given at the 2011 FASDE II Conference in Będlewo, Poland. In the introduction we define Multiple Zeta Values and certain historical remarks are given. Then we present several results on Multiple Zeta Values and, in particular, we introduce certain meromorphic differential equations associated to their generating function. Finally, we make some conclusive remarks on generalisations of Multiple Zeta Values as well as the meromorphic...

Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: $Pu(x,y,z):=(y{\partial}_{x}-z{\partial}_{y})u(x,y,z)=f(x,y,z){\in}_{x,y,z}$, where $P=y{\partial}_{x}-z{\partial}_{y}{:}_{x,y,z}{\to}_{x,y,z}$. For this equation, our aim is to characterize the solvability on ${}_{x,y,z}$ by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.

Dettweiler and Reiter formulated Euler's integral transformation for Fuchsian systems of differential equations and applied it to a definition of the middle convolution. In this paper, we formulate Euler's integral transformation for systems of linear differential equations with irregular singularities. We show by an example that the confluence of singularities is compatible with Euler's integral transformation.

We investigate the growth and fixed points of meromorphic solutions of higher order linear differential equations with meromorphic coefficients and their derivatives. Our results extend the previous results due to Peng and Chen.

Etant donné un système différentiel linéaire de niveau unique quelconque, nous explicitons des formules donnant les multiplicateurs de Stokes en fonction de constantes de connexion dans le plan de Borel, généralisant ainsi les formules obtenues dans l’article Resurgence, Stokes phenomenon and alien derivatives for level-one linear differential systems (M. Loday-Richaud, P. Remy). Pour ce faire, nous nous ramenons à un système de niveaux $\le 1$ par la méthode classique de réduction du rang ; puis, nous...

In this paper we discuss the growth of solutions of the higher order nonhomogeneous linear differential equation $$\begin{array}{cc}& {f}^{\left(k\right)}+{A}_{k-1}{f}^{(k-1)}+\cdots +{A}_{2}{f}^{\text{'}\text{'}}+({D}_{1}\left(z\right)+{A}_{1}\left(z\right){\mathrm{e}}^{az}){f}^{\text{'}}\hfill \\ & +({D}_{0}\left(z\right)+{A}_{0}\left(z\right){\mathrm{e}}^{bz})f=F\phantom{\rule{1.0em}{0ex}}(k\ge 2),\hfill \end{array}$$ where $a$, $b$ are complex constants that satisfy $ab(a-b)\ne 0$ and ${A}_{j}\left(z\right)$$(j=0,1...$

We introduce and discuss a new class of (multivalued analytic) transcendental functions which still share with algebraic functions the property that the number of their isolated zeros can be explicitly counted. On the other hand, this class is sufficiently rich to include all periods (integral of rational forms over algebraic cycles).