It is a general expectation that the refined topological string theory of local Calabi-Yau 3-folds produces quasimodular forms which are solutions of some refined holomorphic anomaly equation. I will present a mathematical derivation that it is indeed the case for the refined topological string on local P^2 in the Nekrasov-Shatashvili limit. The rather indirect proof exploits mathematical incarnations of various string dualities. Along the way, I will present a new way to compute the D4-D2-D0 BPS spectrum of local P^2 in the large volume limit. Work partly joint with Honglu Fan, Shuai Guo, and Longting Wu.