Asymptotic expansions and Stokes multipliers of the confluent hypergeometric function \(\Phi_2\). II: Behaviour near \((\infty,\infty)\) in \(P^1(C)\times P^1(C)\) (Q1127680)

This paper deals with the asymptotic behavior of the confluent hypergeometric function \[ \Phi_2(\beta, \beta', \gamma, x, y) = \sum_{m,n\geq 0} {(\beta)_m (\beta')_m\over (\gamma)_{m+n} (1)_m (1)_n} x^m y^n \] where \((\beta)_m = \Gamma(\beta +m)/\Gamma(m)\). The asymptotic behavior of solutions of the differential system associated with \(\Phi_2\) around \(x=\infty\) and around \(y=\infty\) was given already in the previous paper by the author [\textit{S. Shimomura}, Proc. R. Soc. Edinb., Sect. A 123, No. 6, 1165-1177 (1993; Zbl 0798.33007)]. In this paper, the author gives the Stokes multipliers explicitly by choosing appropriate independent solutions of the system and appropriate sectors around the irregular singular point \((\infty, \infty)\). Then he clarifies the asymptotic behavior of \(\Phi_2\) at each sector.