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Linear independence measures for values of Heine series. (English) Zbl 0653.10031
Let n, N, d be integers with $$0\leq N\leq n<d$$ and let q, $$b_{N+1},...,b_ n\in {\mathbb{C}}$$ satisfy $$0<| q| <1$$ and $$b_ j\neq q$$ 0, $$q^{\pm 1}$$, $$q^{\pm 2}$$,... $$(N+1\leq j\leq n)$$. Let f(z) denote the entire function defined by the series $\sum^{\infty}_{k=0}\frac{q^{dk(k-1)/2}}{Q(q\quad 0)Q(q\quad 1)... Q(q^{k-1})}z\quad k,$ where $$Q(x):=(1-q^{\beta_ 1}x)... (1- q^{\beta_ N}x)(1-b_{N+1}x)... (1-b_ n$$ x) with $$\beta_ 1,...,\beta_ N\in \{1,2,...\}$$. 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 Th. Stihl [Math. Ann. 268, 21-41 (1984; Zbl 0519.10024)].