On a generalized Bessel function of two variables. I (Q1340562)

The author studies the global behavior of functions defined by \[ z_C (x, y)= \int_C \exp(-t^2/ 2-xt- y/t) t^{-\alpha- 1} dt. \] For various \(C\) the integrals provide solutions of a holonomic system on \(\mathbb{P}^1 (\mathbb{C}) \times \mathbb{P}^1 (\mathbb{C})\) of rank 3 with singular loci \(y=0\) of regular type, \(x=\infty\) and \(y=\infty\) of irregular type. For particular \(C\), \(z_C (x, y)\) is considered as an extension of the Bessel function to two variables case. The author gives fundamental solutions for the system by choosing appropriate 3 paths \(C\), computes monodromy matrices and gives asymptotic expansions uniform in \(x\) for the fundamental solutions as \(y\) tends to the irregular singular locus \(y=\infty\). By using these informations, he succeeded in computing Stokes multipliers at \(y= \infty\). It is to be noted that the above functions have already appeared as particular solutions for the nonlinear integrable system, which is an extension of the Painlevé equations to a system of partial differential equations in two variables [\textit{K. Okamoto} and the reviewer, Q. J. Math. 37, 61-80 (1986; Zbl 0597.35114)].