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Nevanlinna theory for the \(q\)-difference operator and meromorphic solutions of \(q\)-difference equations. (English) Zbl 1137.30009

A \(q\)-difference equation of order \(n\) for a meromorphic function \(f\) is an equation involving \(f(z), f(qz),\dots, f(q^nz)\). With the \(q\)-difference operator \(\Delta_q f(z)=f(qz)-f(z)\) such an equation can be viewed as an analogue of a differential equation, with the derivative replaced by the \(q\)-difference operator. The study of such equations is a rich subject with a long history. In the present paper the authors show that if \(f\) has order zero, then \(m(r,f(qz)/f(z))=o(T(r,f))\) on a set of logarithmic density 1. Here \(m(r,\cdot)\) and \(T(r,\cdot)\) are the usual functions of Nevanlinna theory. The above result can be considered as an analogue of the lemma on the logarithmic derivative for the \(q\)-difference operator.
The authors use this result to obtain \(q\)-analogues of the second fundamental theorem of Nevanlinna, the Clunie lemma and a result of A.Z. and V.D. Mohon’ko [Sib. Math. J. 15(1974), 921–934 (1975; Zbl 0305.30029)] about differential equations. Here we mention only one corollary of their version of Nevanlinna’s second fundamental theorem. A value \(a\) is called a \(q\)-Picard exceptional value if all \(a\)-points of \(f\), with at most finitely many exceptions, lie in a set of the form \(\{q^nz: \;n\in \mathbb{N}\cup \{0\}\}\) for some \(z\in \mathbb{C}\). It is shown that a non-constant meromorphic function of order zero has at most two \(q\)-Picard exceptional values.

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
39A13 Difference equations, scaling (\(q\)-differences)
39A70 Difference operators

Citations:

Zbl 0305.30029
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