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

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 ${}_2φ_1$ basic hypergeometric series and the other is given by ${}_2ψ_2$ bilateral basic hypergeometric series.

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