Some results on difference Painlevé IV equations

Abstract

In this paper, we investigate the family of difference Painlevé IV equation$$(w(z+1)+w(z))(w(z)+w(z-1))=R(z,w),$$

where R(z, w) is rational in w and meromorphic in z. If equation assumes an admissible meromorphic solution of hyper-order ${\mathit{\rho }}_2(w)<1$, we give the coefficients relations of all the classification, which are known as the difference Painlevé properties.

Keywords:

• Painlevé difference equation
• meromorphic solution
• iterate

AMS Subject Classifications:

• 39A10
• 30D35

Acknowledgements

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

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the NNSF of China [grant number 11201014], [grant number 11171013], [grant number 11126036]; the Fundamental Research Funds for the Central Universities; CSC and the youth talent programme of Beijing [grant number 29201443].