Existence of Meromorphic Solutions of First-Order Difference Equations

It is shown that if $$\begin{aligned} f(z+1)^n=R(z,f), \end{aligned}$$f(z+1)n=R(z,f),where R(z, f) is rational in f with meromorphic coefficients and $$\deg _f(R(z,f))=n$$degf(R(z,f))=n, has an admissible meromorphic solution, then either f satisfies a difference linear or Riccati equation with meromorphic coefficients, or the equation above can be transformed into one in a list of ten equations with certain meromorphic or algebroid coefficients. In particular, if $$f(z+1)^n=R(z,f)$$f(z+1)n=R(z,f), where the assumption $$\deg _f(R(z,f))=n$$degf(R(z,f))=n has been discarded, has rational coefficients and a transcendental meromorphic solution f of hyper-order $$<1$$<1, then either f satisfies a difference linear or Riccati equation with rational coefficients, or the equation above can be transformed into one in a list of five equations which consists of four difference Fermat equations and one equation which is a special case of the symmetric QRT map. Solutions to all of these equations are presented in terms of Weierstrass or Jacobi elliptic functions, or in terms of meromorphic functions that are solutions to a difference Riccati equation. This provides a natural difference analogue of Steinmetz’ generalization of Malmquist’s theorem.

DOUBLING TROPICAL -DIFFERENCE ANALOGUE OF THE LEMMA ON THE LOGARITHMIC DERIVATIVE

We present a tropical $q$-difference analogue of the lemma on the logarithmic derivative for doubling tropical meromorphic functions.

On the stability of a differential-difference analogue of a two-dimensional problem of integral geometry

A differential-difference analogue of a two-dimensional problem of integral geometry with a weight function is studied. A stability estimate for the problem considered is obtained.

Meromorphic Solutions of Linear Difference Equations with Polynomial Coefficients

We study the growth of the transcendental meromorphic solution f(z) of the linear difference equation:where q(z), p0(z), ..., pn-(z) (n ≥ 1) are polynomials such that p0(z)pn(z) ≢ 0, and obtain some necessary conditions guaranteeing that the order of f(z) satisfies σ(f) ≥ 1 using a difference analogue of the Wiman-Valiron theory. Moreover, we give the form of f(z) with two Borel exceptional values when two of p0(z), ..., pn(z) have the maximal degrees.