  We develop and compare two algorithms for computing first-order right-hand
factors in the ring of linear Mahler operators$\ell_r M^r + \dots + \ell_1 M +
\ell_0$where $\ell_0, \dots, \ell_r$ are polynomials in~$x$ and $Mx = x^b M$
for some integer $b \geq 2$. In other words, we give algorithms for finding all
formal infinite product solutions of linear functional equations$\ell_r(x)
f(x^{b^r}) + \dots + \ell_1(x) f(x^b) + \ell_0(x) f(x) = 0$. The first of our
algorithms is adapted from Petkov\v{s}ek's classical algorithm forthe analogous
problem in the case of linear recurrences. The second one proceeds by computing
a basis of generalized power series solutions of the functional equation and by
using Hermite-Pad{\'e} approximants to detect those linear combinations of the
solutions that correspond to first-order factors. We present implementations of
both algorithms and discuss their use in combination with criteria from the
literature to prove the differential transcendence of power series solutions of
Mahler equations.
