This article continues a previous paper by the authors. Here and there, the two power series F(z) and G(z), first introduced by Dilcher and Stolarsky and related to the so-called Stern polynomials, are studied analytically and arithmetically. More precisely, it is shown that the function field ℂ(z)(F(z),F(z⁴),G(z),G(z⁴)) has transcendence degree 3 over ℂ(z). This main result contains the algebraic independence over ℂ(z) of G(z) and G(z⁴), as well as that of F(z) and F(z⁴). The first statement is due to Adamczewski, whereas the second is our previous main result. Moreover, an arithmetical consequence of the transcendence degree claim is that, for any algebraic α with 0 < |α| < 1, the field ℚ(F(α),F(α⁴),G(α),G(α⁴)) has transcendence degree 3 over ℚ.