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Difference and functional equations Definition

Nevanlinna theory of difference equations

Builds on Meromorphic functionNevanlinna characteristic

Definition

Let c∈ℂ∖{0}c \in \C \setminus \set{0} and let n≥1n \ge 1. Let FF be a polynomial or a rational function in its last n+1n + 1 arguments, with coefficients that are meromorphic functions of zz. A complex difference equation for an unknown meromorphic function ff on ℂ\C is an equation F(z,f(z),f(z+c),…,f(z+nc))=0.F\big(z, f(z), f(z + c), \dots, f(z + nc)\big) = 0. For such a solution ff, write T(r,f)T(r, f) for its Nevanlinna characteristic and m(r,f)m(r, f) for its proximity function. A quantity S(r,f)S(r, f) is small with respect to ff if S(r,f)=o(T(r,f))S(r, f) = o(T(r, f)) as r→∞r \to \infty outside a set of rr of finite logarithmic measure. Nevanlinna theory of difference equations is the study of the meromorphic solutions of such difference equations through TT, mm and the counting function NN. Its basic tools are estimates of the type m(r,f(z+c)/f(z))=S(r,f)m\big(r, f(z + c)/f(z)\big) = S(r, f).

Remarks

Example: f(z)=ezf(z) = e^{z} solves f(z+1)=ef(z)f(z + 1) = e f(z). Here T(r,f)=r/πT(r, f) = r/\pi and f(z+1)/f(z)=ef(z + 1)/f(z) = e is constant, so m(r,f(z+1)/f(z))=1m\big(r, f(z+1)/f(z)\big) = 1, which is small compared with T(r,f)T(r, f).

Non-example: for f(z)=eezf(z) = e^{e^z}, which has infinite order, the quotient f(z+1)/f(z)=e(e−1)ezf(z + 1)/f(z) = e^{(e - 1)e^z} has proximity function comparable to T(r,f)T(r, f), so the small-term estimate fails; this is why the difference lemma requires finite order.

The basic estimate holds for every meromorphic ff of finite order and every c∈ℂc \in \C, outside an exceptional set of finite logarithmic measure.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Nevanlinna_theory
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MSC39A45, 30D35
DefinesNevanlinna theory of difference equations
UsesMeromorphic function, Nevanlinna characteristic

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