Summary.
Local solutions of the functional equation¶¶\( z{^\kappa} \phi \left( z\right) =\sum_{k=1}^nG_k\left( z\right) \phi \left( s_kz \right) +g\left( z\right) \)¶with \( \kappa > 0 \) and \( \left| s_k\right| \gt 1 \) are considered. We prove that the equation is solvable if and only if a certain system of \( \kappa \) conditions on G k (k = 1, 2, ... , n) and g is fulfilled.
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Received: July 7, 1997; revised version: August 3, 1998.
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Mityushev, V. A linear functional equation with a singularity at a fixed point. Aequ. math. 57, 37–44 (1999). https://doi.org/10.1007/s000100050068
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DOI: https://doi.org/10.1007/s000100050068