\(n\)-fold Fourier series expansion in root functions of a differential pencil with \(n\)-fold multiple characteristic (Q311174)

The author considers the following boundary value problem \[ ((d/(dx))-\lambda)^ny=0, \] \[ ((d^{s-1}y)/(dx^{s-1}))|_{x=0}=0,\quad s=1,\dots,n-1, \] \[ ((d^{n-1}y)/(dx^{n-1}))|_{x=1}=\lambda^{n-1}((d^{n-1}y)/(dx^{n-1}))|_{x=0}, \] where \(x\in(0,1)\) and \(\lambda\) is a complex parameter. The Green's function of the problem is constructed by means of the characteristic determinant of the problem. Finally, using two auxiliary results the author obtains an expansion formula in root elements of the boundary value problem.