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Productly linearly independent sequences. (English) Zbl 1363.11073
The notion in the title is defined as follows. $$K$$ sequences $$(a_{k,n})_{n=1}^\infty \; (k=1,\ldots,K)$$ of positive real numbers are said to be productly linearly independent, if the $$K+1$$ numbers $$1$$ and $$\prod_{n=1}^\infty\big(1+1/(a_{k,n}c_n)\big),\; k=1,\ldots,K$$, are $$\mathbb{Q}$$-linearly independent for every sequence $$(c_n)_{n=1}^\infty$$ of positive integers. In the particular case $$K=1$$, a sequence $$(a_n)_{n=1}^\infty$$ of positive real numbers is said to be productly irrational, if $$\prod_{n=1}^\infty\big(1+1/(a_nc_n)\big)$$ is irrational for any sequence $$(c_n)$$ as above. (Note that, in Definition 2, $$\sum_{n=1}^\infty$$ has to be replaced by $$\prod_{n=1}^\infty$$.) For both notions, very technical criteria are given, too cumbersome to be quoted here.
Moreover, the authors show the irrationality of $$(\ast)\!\!: \prod_{n=1}^\infty(1-J^{-2^n})^{-1}$$ for each integer $$J>1$$. It should be pointed out that, using Mahler’s transcendence method, much more can be proved on the infinite product $$\prod_{n\geq1}\big(1+(1-a)z^{d^n}\big)/(1-az^{d^n})$$ defining, for $$(a,d)\in\overline{\mathbb{D}}\times\mathbb{Z}_{\geq2}$$, a function $$P_{a,d}(z)$$ being holomorphic in the open unit disk $$\mathbb{D}$$. Namely, if $$a\in\overline{\mathbb{D}}$$ is algebraic and $$d\in\mathbb{Z}_{\geq2}$$ satisfy $$(a,d)\neq(0,2)$$, then $$P_{a,d}(\alpha)$$ is transcendental for any non-zero algebraic $$\alpha\in\mathbb{D}$$, for which $$(\ast\ast)\!\!:1+(1-a)\alpha^{d^n}\neq0$$ holds for any $$n\in\mathbb{Z}_{\geq1}$$. Note that both conditions, $$(\ast\ast)$$ and $$(a,d)\neq(0,2)$$, are necessary for transcendence, the second one since $$P_{0,2}(z)=(1-z^2)^{-1}$$. Note also that $$P_{1,2}(1/J)$$ is just the authors’ product $$(\ast)$$.
##### MSC:
 11J72 Irrationality; linear independence over a field
##### Keywords:
linear independence; infinite product
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