On transasymptotic expansions of o-minimal germs

Given an o-minimal expansion R₀ of the real ordered field, generated by a generalized quasianalytic class A, we construct an explicit truncation closed ordered differential field embedding of the Hardy field of the expansion R₀, of R₀ by the unrestricted exponential function, into the field T of transseries. We use this to prove some non-definability results. In particular, we show that the restriction to the positive half-line of Euler's Gamma function is not definable in the structure R₀₍^*, , generated by all convergent generalized power series and the exponential function, thus establishing the non-interdefinability of the restrictions to a neighbourhood of + of Euler's Gamma and of the Riemann Zeta function.