2023/02/10 by Yueyang Zhang, Risto Korhonen, Zhang, Yueyang +1
Mathematics · #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #Primary 39A10 #Secondary 30D35 and 39A12
paper · pdf · doi:10.48550/arxiv.2302.05202
openalex publication_date 2023/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently, the present authors used Nevanlinna theory to provide a classification for the Malmquist type difference equations of the form f(z+1)n=R(z,f) (†) that have transcendental meromorphic solutions, where R(z,f) is rational in both arguments. In this paper, we first complete the classification for the case °f(R(z,f))=n of~(†) by identifying a new equation that was left out in our previous work. We will actually derive all the equations in this case based on some new observations on~(†). Then, we study the relations between (†) and its differential counterpart (f')n=R(z,f). We show that most autonomous equations, singled out from~(†) with n=2, have a natural continuum limit to either the differential Riccati equation f'=a+f2 or the differential equation (f')2=a(f2-τ12)(f2-τ22), where a\not=0 and τi are constants such that τ12\not=τ22. The latter second degree differential equation and the symmetric QRT map are derived from each other using the bilinear method and the continuum limit method.