New theory for equations of non-Fuchsian type representation theorem of tree series solution. I (Q1070090)

Consider the higher order equation of non-Fuchsian type \({\mathcal L}(z;d/dz)u(z)=\sum^{t}_{k=0}p_ k(z)z^ k(d/dz)^ ku(z)=0,\) where \(p_ k\) are analytic in a ring-shaped region. Let \[ L(d/dw)=\prod^{t}_{k=1}(d/dw-b_ k),\quad M_ n(d/dw)=a_ n\prod^{m_ n}_{j=1}(d/dw-c_ j)\quad (m_ n\leq t) \] and \(L(d/dw)u(w)=\sum_{n}\exp (nw) M_ n(d/dw)u(w)\), \(n=\pm 1,\pm 2,\ldots\); where \(a_ n\), \(b_ n\) and \(c_ n\) are given constants. Suppose that g(w-w') is the Green function corresponding to \(Lu=\delta (w-w')\) and put \(H(w-w',w')=g(w-w')\sum^{\infty}_{-\infty}\exp (nw') M_ n(d/dw),\) then the non-Fuchsian differential equation of order t is equivalent to a set of t linearly independent integro-differential equations \[ u_ k(w)-\int^{\infty}_{-\infty}H(w-w',w')u_ k(W')dw'=u^ 0_ k(w). \] Here the basic sets of solutions of the non- Fuchsian equation \({\mathcal L}u=0\) and \(Lu=0\) are \(\{u_ k\}\) and \(\{u^ 0_ k\}\) \((k=1,2,\ldots,t)\). The procedure consists of solving the above system of integro-differential equations. For this purpose the two-sided Laplace transform is used.