  The Alcock-Paczynski (AP) effect is a geometrical distortion in
three-dimensional observed galaxy statistics. In anticipation of precision
cosmology based on ongoing and upcoming all-sky galaxy surveys, we build an
efficient method to compute the AP-distorted correlations of galaxy number
density and peculiar velocity fields for any larger angular scale not relying
on the conventionally used plane-parallel (PP) approximation. Here, instead of
the usual Legendre polynomial basis, the correlation functions are decomposed
using tripolar spherical harmonic basis; hence, characteristic angular
dependence due to the wide-angle AP effect can be rigorously captured. By means
of this, we demonstrate the computation of the AP-distorted correlations over
the various scales. Comparing our results with the PP-limit ones, we confirm
that the errors due to the PP approximation become more remarkable as the
visual angle of separation between target galaxies, $\Theta$, enlarges, and
especially for the density auto correlation, the error exceeds $10\%$ when
$\Theta \gtrsim 30^\circ$. This highlights the importance of the analysis
beyond the PP approximation.
