Riccati Solutions of Discrete Painlevé Equations with Weyl Group Symmetry of Type $E_8^{(1)}$

Abstract

We present a special solutions of the discrete Painlevé equations associated with $A_0^{(1)}$, $A_0^{(1)\ast }$ and $A_0^{(1)\ast \ast }$-surface. These solutions can be expressed by solutions of linear difference equations. Here the $A_0^{(1)}$-surface discrete Painlevé equation is the most generic difference equation, as all discrete Painlevé equations can be obtained by its degeneration limit. These special solutions exist when the parameters of the discrete Painlevé equation satisfy a particular constraint. We consider that these special functions belong to the hypergeometric family although they seems to go beyond the known discrete and $q$-discrete hypergeometric functions. We also discuss the degeneration scheme of these solutions.

22 pages

References in corpus (1)

• A $q$-anaolg of the sixth Painlevé equation

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• Elliptic Painlevé equations from next-nearest-neighbor translations on the $E_8^{(1)}$ lattice
• Full-parameter discrete Painlevé systems from non-translational Cremona isometries
• Multiplicative equations related to the affine Weyl group E${}_8$