A meromorphic function is hyper-transcendental if its derivative does not satisfy any not trivial algebraic equation over the field of rational functions. Using Holder’s result on hyper-transcendence of the Gamma function, Mijajlovic and Malesevic established the hyper-transcendence of several other notable functions. We generalized their results to differential algebraic independence in the differential algebra setting using a notion that we called twisted differential maps. In this talk, we will discuss several applications of our results to various series fields and Hardy fields. This is a joint work with Nicholas Pfau.