Existence of Meromorphic Solutions of First-Order Difference Equations

Risto Korhonen,Yueyang Zhang

doi:10.1007/s00365-019-09491-0

Abstract

It is shown that if f(z+1)n=R(z,f),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} f(z+1)^n=R(z,f), \\end{aligned}$$\\end{document}where R(z, f) is rational in f with meromorphic coefficients and deg _f(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), where the assumption deg _f(R(z,f))=n has been discarded, has rational coefficients and a transcendental meromorphic solution f of hyper-order <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.

Highlights

Global existence of large classes of meromorphic solutions is a rare property for a differential equation to have
The existence of globally meromorphic solutions is somewhat more common in the case of difference equations as compared to differential equations. It was shown by Shimomura [23] that the difference equation f (z + 1) = P( f (z)), where P( f (z)) is a polynomial in f (z) with constant coefficients, always has a nontrivial entire solution
Yanagihara [28] showed that the difference equation f (z + 1) = R( f (z)), where R( f (z)) is rational in f (z) having constant coefficients, has a nontrivial meromorphic solution no matter how R is chosen

Summary

Introduction

Global existence of large classes of meromorphic solutions is a rare property for a differential equation to have. Where R(z, f (z)) is rational in both arguments, has a transcendental meromorphic solution of hyper-order strictly less than one, deg f (R(z, f (z))) = 1 and (1.5) reduces into the difference Riccati equation. This is a natural difference analogue of Malmquist’s 1913 result on differential equations. With rational coefficients has a transcendental meromorphic solution f of hyper-order < 1, either f satisfies a difference linear or Riccati equation f (z + 1) = a1(z) f (z) + a2(z),. The proof of our main result, Theorem 2 below, has been split into the three remaining Sects. 4–6

Preliminaries

Extension of the Difference Malmquist Theorem