Abstract
In this paper we deal with complex differential equa- tions of the form f (k) + ak−1(z)f (k−1) + · · · + a1(z)f 0 + a0(z)f = 0 with the coefficients in Fock type space. The relation betweenthe solutions and coefficients in Fock type space is obtained.
Highlights
Motivated by the work in [6], [7] and [8], we will study complex differential equations of the form (1) f (k) + ak−1(z)f (k−1) + · · · + a1(z)f ′ + a0(z)f = 0 where the coefficients are entire functions
The Bargmann-Fock space is the Hilbert space of entire functions equipped with the inner product
This space has been studied by many authors and it is rooted from mathematical problems of relativistic physics or from quantum optics
Summary
√ normed by f α = < f, f >α, where α(r) is a nonnegative and nondecreasing function of r. We will consider the growth relation between the coefficients and the solutions of (1). (ii)Suppose that all solutions of (1) belong to the Fock-type space Feα, find out whether all of the coefficients aj(z), j = 0, ..., k − 1 belong to the Fock-type space Fα.
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