Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 090, 23 pages      arXiv:1301.2401      https://doi.org/10.3842/SIGMA.2014.090

Hypergeometric Solutions of the $A_4^{(1)}$-Surface $q$-Painlevé IV Equation

Nobutaka Nakazono
School of Mathematics and Statistics, The University of Sydney, New South Wales 2006, Australia

Received June 06, 2013, in final form August 14, 2014; Published online August 22, 2014

Abstract
We consider a $q$-Painlevé IV equation which is the $A_4^{(1)}$-surface type in the Sakai's classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric functions. Two of them are expressed by ${}_2\varphi_1$ basic hypergeometric series and the other is given by ${}_2\psi_2$ bilateral basic hypergeometric series.

Key words: $q$-Painlevé equation; basic hypergeometric function; affine Weyl group; $\tau$-function; projective reduction; orthogonal polynomial.

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