Construction of asymptotics of solutions of differential equations with cusp-type degeneration in the coefficients in the case of multiple roots of the highest-order symbol (Q1649465)

Consider differential equations of the form \[ H\Biggl(r,-r^2{d\over dr}\Biggr)\,u= 0,\tag{\(*\)} \] where \(H(r,p)= \sum^n_{i=1} b_i p^i\), and \(b_i\) are holomorphic functions in a neighborhood of zero. The authors represent \(H(r,p)\) in the form \[ H(r,p)= \sum^n_{i=0} b^0_i p^i+ r\sum^n_{i=0} b^1_i p^i+ \sum^n_{i=0}\, \Biggl(\sum^\infty_{j=2} b^j_i r^j\Biggr) p^i \] and assume that \(\sum^n_{i=0} b^0_i p^i= 0\) has multiple roots and that these roots do not coincide with the roots of \(\sum^n_{i=0} b^1_i p^i=0\). Under these assumptions they determine the asymptotics of the solutions of \((*)\).