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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)].
Reviewer: M.Katsurada

MSC:
11J81 Transcendence (general theory)
11J85 Algebraic independence; Gel’fond’s method
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References:
[1] Katsurada, M.: Linear independence measures for certain numbers. Result. Math.14, 318-329 (1988) · Zbl 0659.10039
[2] Mahler, K.: Perfect systems. Compos. Math.19, 95-166 (1968) · Zbl 0168.31303
[3] Stihl, T.: Arithmetische Eigenschaften spezieller Heinescher Reihen. Math. Ann.268, 21-41 (1984) · Zbl 0533.10031
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