On the value distribution theory for entire solutions of second-order linear differential equations.

*(English)*Zbl 0545.30022There has been extensive research (including work of W. Hayman, J. Clunie, L. Sons, E. Mues, and G. Frank and their students) in producing certain classes of polynomials \(P(z,u_ 0,...,u_ n)\) in the indeterminates \(u_ 0,...,u_ n\), having meromorphic coefficients, which have the following property: For any entire transcendental function f(z) which grows faster than the coefficients of P, the function \(h_ f(z)=P(z,f(z),f'(z),...,f^{(n)}(z))\) must have infinitely many zeros. In some cases, it has been shown that the counting function for the distinct zeros of \(h_ f(z)\) cannot grow slower than the growth of f(z).

In the present paper, we show that in general, for a solution f(z) of a linear differential equation \(w''+A(z)w=0,\) where A(z) is an arbitrary entire function, the above results can be fully completed in a very precise way, namely by completely determining which polynomials \(P(z,u_ 0,...,u_ n)\), for various classes of meromorphic coefficients of slower growth than f(z), have the property that either \(h_ f(z)\equiv 0\) or the counting function for the distinct zeros of \(h_ f(z)\) grows at most like the coefficients of P. The actual determination of these exceptional polynomials depends on the distribution of zeros of all the solutions of the differential equation \(w''+A(z)w=0.\) In addition, general examples are constructed which fully illustrate the results. Finally, we mention that the class of functions treated includes some of the special functions which arise in applications, such as Airy functions (where \(A(z)=-z)\), and Mathieu functions (where \(A(z)=a+b(Cos(2z)),\) for complex constants a and b with \(b\neq 0)\).

In the present paper, we show that in general, for a solution f(z) of a linear differential equation \(w''+A(z)w=0,\) where A(z) is an arbitrary entire function, the above results can be fully completed in a very precise way, namely by completely determining which polynomials \(P(z,u_ 0,...,u_ n)\), for various classes of meromorphic coefficients of slower growth than f(z), have the property that either \(h_ f(z)\equiv 0\) or the counting function for the distinct zeros of \(h_ f(z)\) grows at most like the coefficients of P. The actual determination of these exceptional polynomials depends on the distribution of zeros of all the solutions of the differential equation \(w''+A(z)w=0.\) In addition, general examples are constructed which fully illustrate the results. Finally, we mention that the class of functions treated includes some of the special functions which arise in applications, such as Airy functions (where \(A(z)=-z)\), and Mathieu functions (where \(A(z)=a+b(Cos(2z)),\) for complex constants a and b with \(b\neq 0)\).