On the question of whether f″+ e−zf′ + B(z)f = 0 can admit a solution f ≢ 0 of finite order

Abstract

SynopsisWe show that if B(z) is either (i) a transcendental entire function with order (B)≠1, or (ii) a polynomial of odd degree, then every solution f≠0 to the equation f″ + e−zf′ + B(z)f = 0 has infinite order. We obtain a partial result in the case when B(z) is an even degree polynomial. Our method of proof and lemmas for case (i) of the above result have independent interest.