Linear independence measures for values of Heine series (Q1107569)

Let \(n, N, d\) be integers with \(0\leq N\leq n<d\) and let \(q\), \(b_{N+1},\ldots,b_n\in\mathbb{C}\) satisfy \(0<| q| <1\) and \(b_j\neq q^0\), \(q^{\pm 1}\), \(q^{\pm 2},\ldots\) \((N+1\leq j\leq n)\). Let \(f(z)\) denote the entire function defined by the series \[ \sum^{\infty}_{k=0}\frac{q^{dk(k-1)/2}}{Q(q^0)Q(q^1) \cdots Q(q^{k-1})}z^k, \] where \(Q(x):=(1-q^{\beta_1}x) \cdots (1- q^{\beta_N}x)(1-b_{N+1}x)\cdots(1-b_n x)\) with \(\beta_1,\ldots,\beta_N\in \{1,2,\ldots\}\). Then using the method of Padé approximation, we obtain the linear independence measures for the values of \(f(z)\) as well as its derivatives of any order. It is an improvement of the earlier result of \textit{Th. Stihl} [Math. Ann. 268, 21--41 (1984; Zbl 0519.10024)].