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ν=0 q−ν(ν+1)/2 z ν, |q |> 1, be the Chakalov (or Tschakaloff) series. The authors prove the linear independence of the numbers 1, T q t 1(α) and T q t 2(α) over Q, where t1, t2 are two different positive integers, and the numbers q ∈ Z � {0, ±1}, α ∈ Q � {0} are multiplicatively independent. Moreover, they also give the following result: Let I be any imaginary quadratic field and q ∈ ZI, |q |> 1. Suppose that t1,..., tl are pairwise different positive integers, and that the numbers α ∈ I � {0} are multiplicatively independent with q. Then the numbers 1, T q t k(α) (k = 1,..., l) are linear independent over I. Furthermore, if the numbers αk (k = 1,..., l) ∈ I � {0} are multiplicatively independent with q, then the numbers 1, T q t k(αk) (k = 1,..., l) are linear independent over I. Only outlines of the proofs are given

Year: 2013

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