COMPLETELY REGULAR GROWTH SOLUTIONS OF SECOND ORDER COMPLEX LINEAR DIFFERENTIAL EQUATIONS

摘要

If A(z) and B(z) are transcendental entire functions, then all solutions of the differential equationf'' + A(z)f' + B(z)f = 0 are entire and typically of infinite order. Simple examples show that finite order solutions are alsopossible. Assuming that A(z) and B(z) are of completely regular growth, Gol'dberg-Ostrovskii-Petrenkoasked whether all solutions of finite order are of completely regular growth also. This problemremains unsolved, but several aspects of the problem are addressed. Exponential polynomials form animportant subclass of functions of completely regular growth, and they are known to always satisfycertain linear differential equations. Hence cases where the coefficientsand/or the solutions are exponential polynomials are also discussed.