Algebraic independence of some Lambert series

1] proved that if Y k0 (1 0 z F k) = X k0 "(k)z k ; then "(k) = 0 or 61 for any k 0. Tamura [7] generalized this result by proving the following theorem: Let fR k g k0 be a linear recurrence of positive integers satisfying R k+n = R k+n01 + 1 1 1 + R k (k 0) with n 2 and let P(z) = Y k0 (1 0 z R k) = X k0 "(k)z k : Then, if n is even, f"(k) j k 0g is a nite set; if in addition R k = 2 k (0 k n01), "(k) = 0 or 61 for any k 0. He also showed that P (g 01) is irrational for any integer g with jgj 2. In the same paper, he studied a Lambert type series 2(z) = and proved, using its continued fraction expansion, that 2(g 01) is irrational for any integer g 2. It is conjectured in [7] that P () and 2() are transcendental for any algebraic number with 0 < jj < 1. We note that the transcendency of P (), and even the algebraic independence of the values of P(z) at distinct algebraic numbers, can be deduced from Theorem 5 in [9] : Let 1 ; 1 1 1 ; r be algebraic numbers with 1