  In this paper, we deal with uncertainty quantification for the random
Legendre differential equation, with input coefficient $A$ and initial
conditions $X_0$ and $X_1$. In a previous study [Calbo G. et al, Comput. Math.
Appl., 61(9), 2782--2792 (2011)], a mean square convergent power series
solution on $(-1/e,1/e)$ was constructed, under the assumptions of mean fourth
integrability of $X_0$ and $X_1$, independence, and at most exponential growth
of the absolute moments of $A$. In this paper, we relax these conditions to
construct an $\mathrm{L}^p$ solution ($1\leq p\leq\infty$) to the random
Legendre differential equation on the whole domain $(-1,1)$, as in its
deterministic counterpart. Our hypotheses assume no independence and less
integrability of $X_0$ and $X_1$. Moreover, the growth condition on the moments
of $A$ is characterized by the boundedness of $A$, which simplifies the proofs
significantly. We also provide approximations of the expectation and variance
of the response process. The numerical experiments show the wide applicability
of our findings. A comparison with Monte Carlo simulations and gPC expansions
is performed.
