Hypergeometric solutions to the q-Painlevé equations
نویسنده
چکیده
It is well-known that the continuous Painlevé equations PJ (J=II,. . .,VI) admit particular solutions expressible in terms of various hypergeometric functions. The coalescence cascade of hypergeometric functions, from the Gauss hypergeometric function to the Airy function, corresponds to that of Painlevé equations, from PVI to PII [1]. The similar situation is expected for the discrete Painlevé equations. The discrete Painlevé equations and their solutions have been studied for many years from various view points. In particular, Sakai [2] gave a natural framework of discrete Painlevé equations by means of geometry of rational surfaces. Among the 22 types in Sakai’s classification, there are ten types of q-Painlevé equations corresponding to the following degeneration diagram of type of affine Weyl groups:
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