  The cross-correlation of the Ly-alpha forest and redshifted 21-cm emission
has recently been proposed as an observational tool for mapping out the
large-scale structures in the post-reionization era z < 6. This has a
significant advantage as the problems of continuum subtraction and foreground
removal are expected to be considerably less severe in comparison to the
respective auto-correlation signals. Further, the effect of discrete quasar
sampling is less severe for the cross-correlation in comparison to the Ly-alpha
forest auto-correlation signal. In this paper we explore the possibility of
using the cross-correlation signal to detect the baryon acoustic oscillation
(BAO). To this end, we have developed a theoretical formalism to calculate the
expected cross-correlation signal and its variance. We have used this to
predict the expected signal, and estimate the range of observational parameters
where a detection is possible.
  For the Ly-$\alpha$ forest, we have considered BOSS and BIGBOSS which are
expected have a quasar density of 16 deg^{-2} and 64 deg^{-2} respectively. A
radio interferometric array that covers the redshift range z=2 to 3 using
antennas of size 2 m * 2 m, is well suited for the 21-cm observations. It is
required to observe 25 independent fields of view, which corresponds to the
entire angular extent of BOSS. We find that it is necessary to achieve a noise
level of 1.1 * 10^{-5} K^2 and 6.25 * 10^{-6} mK^2 per field of view in the
21-cm observations to detect the angular and radial BAO respectively with BOSS.
The corresponding figures are 3.3 * 10^{-5} mK^2 and 1.7 * 10^{-5} mK^2 for
BIGBOSS. Four to five independent radio interferometric arrays, each containing
400 antennas uniformly sampling all the baselines within 50 m will be able to
carry out these observations in the span of a few years.
