Let be the Fibonacci sequence. Motivated by the identity , Erdös and Graham asked whether is irrational for any sequence of positive integers with . We resolve the transcendence counterpart of their question; as a special case of our main theorem, we have that is transcendental when . The bound is best possible thanks to the identity at the beginning. This paper provides a new way to apply the subspace theorem to obtain transcendence results and extends previous nontrivial results obtainable by only Mahler’s method for special sequences of the form .
Khoa Dang Nguyen. "Transcendental series of reciprocals of Fibonacci and Lucas numbers." Algebra Number Theory 16 (7) 1627 - 1654, 2022. https://doi.org/10.2140/ant.2022.16.1627