Abstract
Consider the homogeneous linear differential equation where the coefficients aj(z) are entire functions. Then every solution w(z)≢0 of this equation (*) is an entire function. In this paper we give the necessary and sufficient conditions that n-1 linearly independent entire functions satisfy a differential equation (*) of order n. Especially we prove the following theorem: Given k linearly independent entire functions g1(z), g2(z),..., gk(z). This functions are solutions of a differential equation (*) if and only if there exists an integer M<∞ such that for any linear combination g(z)=C1g1(z)+...+ckgk(z)≢O this number M is an upper bound for the multiplicity of the zeros of g(z). Then holds n≧k.
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