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Painlevé III and V Types Differential Difference Equations

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Abstract

In this paper, we show that if the equations

$$\begin{aligned} w(z+1)w(z-1)+a(z)\frac{w'(z)}{w(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{aligned}$$

and

$$\begin{aligned} (w(z)w(z+1)-1)(w(z)w(z-1)-1)+a(z)\frac{w'(z)}{w(z)}=\frac{P(z,w(z))}{Q(z,w(z))}, \end{aligned}$$

where a(z) is rational, P(z, w) and Q(z, w) are coprime polynomials of w(z) with rational functions coefficients, have a non-rational meromorphic solution with hyper-order less than one, then the degrees of the numerator and denominator on the right sides of the equations have to meet certain conditions.

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Acknowledgements

The authors would like to thank the referee for his or her valuable suggestions for the present paper.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beihang University, Bei**g, 100191, People’s Republic of China

    Yishuo Du & Jilong Zhang

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  1. Yishuo Du
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  2. Jilong Zhang
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Correspondence to Jilong Zhang.

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Communicated by Ilpo Laine.

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This research was supported by the NNSF of China, No. 11201014. This research was also supported by the youth talent program of Bei**g, No. 29201443.

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Du, Y., Zhang, J. Painlevé III and V Types Differential Difference Equations. Comput. Methods Funct. Theory 23, 327–345 (2023). https://doi.org/10.1007/s40315-022-00442-8

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