Finite order solutions of second order linear differential equations

Gundersen, Gary G.

doi:10.1090/S0002-9947-1988-0920167-5

Finite order solutions of second order linear differential equations
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Trans. Amer. Math. Soc. 305 (1988), 415-429 Request permission

Abstract:

We consider the differential equation $f'' + A(z)f’ + B(z)f = 0$ where $A(z)$ and $B(z)$ are entire functions. We will find conditions on $A(z)$ and $B(z)$ which will guarantee that every solution $f\not \equiv 0$ of the equation will have infinite order. We will also find conditions on $A(z)$ and $B(z)$ which will guarantee that any finite order solution $f\not \equiv 0$ of the equation will not have zero as a Borel exceptional value. We will also show that if $A(z)$ and $B(z)$ satisfy certain growth conditions, then any finite order solution of the equation will satisfy certain other growth conditions. Related results are also proven. Several examples are given to complement the theory.

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