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Representation of the solution of a homogeneous convolution equation in the form of a series. (English. Russian original) Zbl 0741.30021
Math. Notes 49, No. 3, 259-262 (1991); translation from Mat. Zametki 49, No. 3, 46-51 (1991).
Let \(f_ j (j=0,1,2,\ldots)\) be entire functions of exponential type and let \(f_ 0(z)=\prod^ \infty_{j=1}f_ j(z).\) The main result of the paper contains the non-improvable conditions which provide the representability of any entire solution of the equation \(f_ 0(d/dz)y_ 0(z)=0\) in the form \(y_ 0(z)=\sum^ \infty_{j=1}y_ j(z),\) where \(y_ j(z)\) are entire solutions of the equations \(f_ j(d/dz)y_ j(z)=0 (j=1,2,\ldots)\).
MSC:
30D10 Representations of entire functions of one complex variable by series and integrals
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
47A50 Equations and inequalities involving linear operators, with vector unknowns
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