  Let $ \{\varphi_i(z;\alpha)\}_{i=0}^\infty $, corresponding to $
\alpha\in(-1,1) $, be orthonormal Geronimus polynomials. We study asymptotic
behavior of the expected number of real zeros, say $ \mathbb E_n(\alpha) $, of
random polynomials \[ P_n(z) := \sum_{i=0}^n\eta_i\varphi_i(z;\alpha), \] where
$ \eta_0,\dots,\eta_n $ are i.i.d. standard Gaussian random variables. When $
\alpha=0 $, $ \varphi_i(z;0)=z^i $ and $ P_n(z)$ are called Kac polynomials. In
this case it was shown by Wilkins that $ \mathbb E_n(0)$ admits an asymptotic
expansion of the form \[ \mathbb E_n(0) \sim \frac2\pi\log(n+1) +
\sum_{p=0}^\infty A_p(n+1)^{-p} \] (Kac himself obtained the leading term of
this expansion). In this work we obtain a similar expansion of $ \mathbb
E(\alpha) $ for $ \alpha\neq 0 $. As it turns out, the leading term of the
asymptotics in this case is $ (1/\pi)\log(n+1) $.
