Transcendence of reciprocal sums of binary recurrences

Let {R _n } _n≥0 be a binary linear recurrence defined by R _n+2 = A R _n+1 + B R _n (n ≥ 0), where A, B, R ₀, R ₁ are integers and Δ = A ² + 4B > 0. We give necessary and sufficient conditions for the transcendence of the numbers

?k 3 0￠fracakRrk+b,sum_{kgeq 0}{}^{prime}frac{a_k}{R_{r^k}+b},
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