Hypergeometric Solutions of the $A_4^{(1)}$-Surface $q$-Painlev\'e IV Equation

We consider a $q$-Painlev'e 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 ${}_2{\phi }_1$ basic hypergeometric series and the other is given by ${}_2{\psi }_2$ bilateral basic hypergeometric series.