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Wiman-Valiron method for difference equations. (English) Zbl 1070.39002

The authors study the solutions of the linear difference equation \[ a_p(z)\,\Delta^pf(z)+ \cdots + a_1(z)\,\Delta f(z)+ a_0(z)\,f(z) =0, \] where \(a_j(z)=\sum_{k=0}^{A_j}{a_k^{(j)}z^k}\), \(A_j=\deg a_j(z)\) and \(a_p(z) \not \equiv 0\). They consider an analogue to the Wiman-Valiron theory rewriting power series of an entire function \(f\) of order less than \(1/2\) into binomial series.

MSC:

39A10 Additive difference equations
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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References:

[1] Vorlesungen über Differenzenrechnung, Chelsea Publ. (1954)
[2] W. Gruyter (1992)
[3] DOI: 10.1007/BF02795340 · Zbl 0101.05302
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[5] Neuere Untersuchungen über eindeutige analytische Funktionen (1955) · Zbl 0067.05501
[6] DOI: 10.1090/S0002-9947-98-02080-7 · Zbl 0893.34003
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[8] DOI: 10.1007/s00010-002-8012-x · Zbl 1009.39022
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[10] DOI: 10.4153/CMB-1974-064-0 · Zbl 0314.30021
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