Natalya Zheltukhina

Asymptotic zero distribution of sections and tails of Mittag-Leffler functions

Abstract We study the asymptotic (as n →∞) zero distribution of (1− λ ) s n ( z )− λt n +1 ( z ), where λ∈ C , s n is n th section, t n is n th tail of the power series of Mittag–Leffler function E 1/ ρ of order ρ >1. Our results generalize the results by Edrei, Saff and Varga for the case λ =0. To cite this article: N. Zheltukhina, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 133–138.

On Darboux-integrable semi-discrete chains

A differential-difference equation with unknown t(n, x) depending on the continuous and discrete variables x and n is studied. We call an equation of such kind Darboux integrable if there exist two functions (called integrals) F and I of a finite number of dynamical variables such that DxF = 0 and DI = I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n) = p(n + 1). It is proved that the integrals can be brought to some canonical form. A method of construction of an explicit formula for a general solution to Darboux-integrable chains is discussed and such solutions are found for a class of chains.

On the classification of Darboux integrable chains

We study a differential-difference equation of the form tx(n+1)=f(t(n),t(n+1),tx(n)) with unknown t=t(n,x) depending on x and n. The equation is called a Darboux integrable if there exist functions F (called an x-integral) and I (called an n-integral), both of a finite number of variables x,t(n),t(n±1),t(n±2),…,tx(n),txx(n),…, such that DxF=0 and DI=I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n)=p(n+1). The Darboux integrability property is reformulated in terms of characteristic Lie algebras that give an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial x-integrals is given in the case when the function f is of the special form f(x,y,z)=z+d(x,y).

Complete list of Darboux integrable chains of the form t1x = tx +d (t, t1)

We study differential-difference equation (d/dx)t(n+1,x)=f(t(n,x),t(n+1,x),(d/dx)t(n,x)) with unknown t(n,x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, {t(n+k,x)}k=−∞∞, {(dk/dxk)t(n,x)}k=1∞, such that DxF=0 and DI=I, where Dx is the operator of total differentiation with respect to x and D is the shift operator: Dp(n)=p(n+1). Reformulation of Darboux integrability in terms of finiteness of two characteristic Lie algebras gives an effective tool for classification of integrable equations. The complete list of Darboux integrable equations is given in the case when the function f is of the special form f(u,v,w)=w+g(u,v).

Discretization of hyperbolic type Darboux integrable equations preserving integrability

A method of integrable discretization of the Liouville type nonlinear partial differential equations based on integrals is suggested. New examples of the discrete Liouville type models are presented.

On existence of an x-integral for a semi-discrete chain of hyperbolic type

A class of semi-discrete chains of the form

Discretization of Liouville type nonautonomous equations preserving integrals

t_{1x} = f(x, t, t_1, t_x)

Semi-discrete hyperbolic equations admitting five dimensional characteristic x-ring

is considered. For the given chains easily verifiable conditions for existence of x-integral of minimal order 4 are obtained.

On the discretization of Laine equations

The problem of constructing semi-discrete integrable analogues of the Liouville type integrable PDE is discussed. We call the semi-discrete equation a discretization of the Liouville type PDE if these two equations have a common integral. For the Liouville type integrable equations from the well-known Goursat list for which the integrals of minimal order are of the order less than or equal to two we presented a list of corresponding semi-discrete versions. The list contains new examples of non-autonomous Darboux integrable chains.

On Power Series having Sections with Multiply Positive Coefficients

The necessary and sufficient conditions for a hyperbolic semi-discrete equation to have five dimensional characteristic x-ring are derived. For any given chain, the derived conditions are easily verifiable by straightforward calculations.