Laplace approximation for Bessel functions of matrix argument

We derive Laplace approximations to three functions of matrix argument which arise in statistics and elsewhere: matrix Bessel Aν; matrix Bessel Bν; and the type II confluent hypergeometric function of matrix argument, Ψ. We examine the theoretical and numerical properties of the approximations. On the theoretical side, it is shown that the Laplace approximations to Aν, Bν and Ψ given here, together with the Laplace approximations to the matrix argument functions 1F1 and 2F1 presented in Butler and Wood (Laplace approximations to hyper-geometric functions with matrix argument, Ann. Statist. (2002)), satisfy all the important confluence relations and symmetry relations enjoyed by the original functions.