On a generalized Bessel function of two variables. II: Case of coalescing saddle points (Q1281712)

This paper concerns a study of the global behavior of solutions of the system \[ \partial_x^2 u= x \partial_x u - y \partial _y u - \alpha u, \quad \partial_x\partial_y u= u,\quad y\partial_y^2 u= - \partial_x u - (\alpha + 1) \partial _y u + u, \] which is a holonomic system on \( {\mathbb C}^2\) with regular singularity along \( y =0,\) irregular singularity along \( x = \infty\) and \( y = \infty\) having \(3\) independent solutions at nonsingular points. The system admits solutions given by contour integrals \[ u(x,y) = \int_C \exp\left(- \frac{t^2}{2} - xt - \frac{y}{t}\right) t^{-\alpha-1} dt \] with appropriate contours on the \( t \)-space \( {\mathbb C} \setminus \{0\},\) which the author calls a generalized Bessel function of two variables. The main results of this paper are (1): the determination of asymptotic expansions of fundamental solutions of the system along the irregular singularity \( x =0\) and (2) the computation of the Stokes multipliers around this singularity. The singular loci \( y = 0\) and \( y = \infty\) have been already studied in the author's previous paper [J. Math. Anal. Appl. 187, No. 2, 468-464 (1994; Zbl 0844.33009)].