Hypergeometric Solutions of the A (1) 4-Surface q-Painleve ́ IV Equation

Abstract. We consider a q-Painleve ́ IV equation which is the A (1) 4-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 2ϕ1 basic hypergeometric series and the other is given by 2ψ2 bilateral basic hypergeo-metric series. Key words: q-Painleve ́ equation; basic hypergeometric function; affine Weyl group; τ-func-tion; projective reduction; orthogonal polynomia