  This paper derives new integral representations for products of two parabolic
cylinder functions. In particular, expressions are obtained for
D_{nu}(x)D_{mu}(y), with x>0 and y>0, that allow for different orders and
arguments in the two parabolic cylinder functions. Also, two integral
representations are obtained for D_{nu}(-x)D_{mu}(y) by employing the
connection between the parabolic cylinder function and the Kummer confluent
hypergeometric function. The integral representations are specialized for
products of two complementary error functions and of two modified Bessel
functions of the second kind of order 1/4, as well as for the product of a
parabolic cylinder function and a modified Bessel function of the first kind of
order 1/4.
