Calculation of the Stokes' multipliers for a polynomial system of rank 1 having distinct eigenvalues at infinity (Q1310526)

R. Schafke (in a recent publication) showed that the connection data of the system \(zx'=(zA_ 0+A_ 1)x\) can be computed from certain connection constants of the so-called hypergeometric system \((A_ 0-t) dy/dt=(A_ 1-s)y\) where \(A_ 0\), and \(A_ 1\) are \(n \times n\) matrices, \(t\) and \(s\) are complex parameters, or it can also be obtained from the difference equation \((A_ 0-t) f(s,t)=(s-A_ 1) f(s+1,t)\). The author here uses the above results and the techniques to establish this theorem: The system \(G(s+1) (s+B_ 1-s^{-1} ab^ T)=G(s) B_ 0\), \(B_ 0=\text{diag} [\lambda_ n-\lambda_ 1\), \(\lambda_ n- \lambda_ 2, \dots, \lambda_ n -\lambda_{n-1}]\) has a solution \(G(s)\) uniquely determined and it is meromorphic throughout the complex \(s\)-plane with possible poles at the points \(s=0,-1,-2,\dots\).