On the values of certain \(q\)-hypergeometric series (Q2710111)

In a previous paper, the authors gave an irrationality theorem for values of certain hypergeometric functions satisfying the Poincaré equation (*) \(\alpha z^sf(z)=P(z)f(qz)-P(z)\) and defined over an algebraic number field \(K\) [Acta Arith. 99, 389--407 (2001; Zbl 0984.11037)]. The exceptional set, \({\mathcal E}_q(s,P)\), of \(\alpha\) in \(K\) for which both \(\alpha\) and the function value \(f(\alpha)\) are both in \(K\) is the set of \(\alpha\) for which the functional equation (*) has a solution \(f(z)\) in \(K[z]\). Here, \(s\geq 2\) is a positive integer, \(q\) is a nonzero integer in \(K\), \(P(z)=\sum_{i=0}^s a_is^i\neq a_sz^s+a_0\) is in \(K[z]\) and \(a_sa_0\neq 0\). The authors show that \({\mathcal E}_q(s,P)=\{0\}\) in many cases, but give examples to show that this is not always the case.NEWLINENEWLINENEWLINEIf (*) has a polynomial solution, then its coefficients satisfy a system of linear equations which leads firstly to the conclusion that \(\alpha=a_sq^n\) and then to the vanishing of certain determinants. These conditions can only be satisfied by finitely many \(q\) and, in special cases, the exceptional cases to be enumerated.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].