Subnormal solutions of second order linear differential equations with periodic coefficients
A very complete analysis is conducted for the second order linear differential equation \[ f''+ P(e^ z)f'+ Q(e^ z)f= R_ 1(e^ z)+ R_ 2(e^{-z}),\leqno{(*)} \] where \(P(z)\), \(Q(z)\), \(R_ 1(z)\), and \(R_ 2(z)\) are polynomials in \(z\), such that \(P(z)\) and \(Q(z)\) are not both constant. We say \(f\) is a subnormal solution of equation \((*)\) iff \(f\) satisfies \((*)\) and moreover satisfies the \(R\). Nevanlinna's condition, \[ \varlimsup_{r\to\infty} {\log T(r,f)\over r}= 0. \] It is well-known that the subnormal solution to the homogeneous problem, namely, \(R_ 1(e^ z)= 0\) and \(R_ 2(e^{-z})= 0\) is given by the function \(f(z)= e^{cz}(a_ 0+ a_ 1 e^ z+\cdots+ a_ m e^{mz})\), where \(m\geq 0\) is an integer and \(c,a_ 0,a_ 1,\dots,a_ m\) are constants with \(a_ 0\neq 0\) and \(a_ m\neq 0\). Four remarkable theorems are then stated with very excellent detailed proofs included in latter sections. They relate the degree of the polynomials, \(P\), \(Q\), \(R_ 1\) and \(R_ 2\) and this effect is then transposed to the solutions of the problem. For example if \(\deg P=\deg Q\geq 1\) then the subnormal solution \(f\) to equation \((*)\) must have one of the following two forms: \(f(z)= S_ 1(e^ z)+ S_ 2(e^{-z})+ e^{bz} S_ 3(e^ z)\), where \(S_ 1(z)\), \(S_ 2(z)\), and \(S_ 3(z)\) are polynomials in \(z\), and \(b\) is constant; or \[ f(z)= e^{nz}\left\{cz \sum^ m_{k=0} a_ k e^{- kz}+ S_ 1(e^ z)+ S_ 2(e^{-z})\right\}, \] where \(m\geq 0\) and \(n\) are integers, \(c,a_ 0,a_ 1,\dots,a_ m\) are constants with \(c\neq 0\), \(a_ 0\neq 0\) and \(a_ m\neq 0\). Again \(S_ 1(z)\) and \(S_ 2(z)\) are polynomials in \(z\). All proofs of the theorems are given in significant detail and several examples illustrating the results are also included.
second order linear differential equation
scientific article; zbMATH DE number 604854
scientific article