Connection formulae for solutions of a system of partial differential equations associated with the confluent hypergeometric function \(\Phi_ 2\) (Q1924889)

The confluent hypergeometric function \[ \Phi_2 (\beta, \beta', \gamma, x,y) = \sum_{m,n\geq 0} {(\beta)_m (\beta')_n \over (\gamma)_{m+n} (1)_m(1)_n} x^my^n \tag{1} \] with \((\beta)_m = \Gamma (\beta+m)/ \Gamma (\beta)\) satisfies a system of partial differential equations (2) \(xz_{xx} + yz_{xy} + (\gamma-x) z_x- \beta z=0\), \(yz_{yy} + xz_{xy} + (\gamma-y) z_y-\beta'z = 0\), which possesses the singular loci \(x=0\), \(y=0\), \(x-y=0\) of regular type and \(x=\infty\), \(y=\infty\) of irregular type. The solutions of (2) constitute a three-dimensional vector space over \(\mathbb{C}\). Near each singular locus of regular type, system (2) admits a triplet of linearly independent solutions expanded into convergent series [\textit{A. Erdélyi}, Proc. R. Soc. Edinb. 59, 224-241 (1939; Zbl 0027.39903)], ibid. 60, 344-361 (1940; Zbl 0063.01258)], and near the singular loci of irregular type, the author examined the asymptotic behaviour of suitably chosen linearly independent solutions [the author, Proc. R. Soc. Edinb. Sect. A 123, No. 6, 1165-1177 (1993; Zbl 0798.33007)]. In this paper, connection formulae for these solutions are calculated.