The discrete second Painlevé equation dPII is mapped to the second Painlevé equation PII by its continuous limit, and then, as shown by Kajiwara et al., a rational solution of dPII also reduces to that of PII. In this paper, regarding dPII as a difference equation, we present a certain asymptotic solution that reduces to a triply-truncated solution of PII in this continuous limit. In a special case our solution corresponds to a rational one of dPII. Furthermore we show the existence of families of solutions having sequential limits to truncated solutions of PII.
Shun SHIMOMURA. "Continuous limit of the difference second Painlevé equation and its asymptotic solutions." J. Math. Soc. Japan 64 (3) 733 - 781, July, 2012. https://doi.org/10.2969/jmsj/06430733