Wiman-Valiron method for difference equations

Abstract

Abstract. Let f (z) be an entire function of order less than 1/2. We consider an analogue of the Wiman-Valiron theory rewriting power series of f (z) into binomial series. As an application, it is shown that if a transcendental entire solution f (z) of a linear difference equation is of order χ < 1/2, then we have log M (r, f ) = Lr χ (1 + o (1)) with a constant L > 0.

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

“…Recently, there has been renewed interests in difference (discrete) equations in the complex plane C ( [1], [8], [17], [22], [26], [37]; see also [6]). In particular, and most noticeably, is the proposal by Ablowitz, Halburd and Herbst [1] to use the notion of order of growth of meromorphic functions in the sense of classical Nevanlinna theory [15] as a detector of integrability (i.e., solvability) of second order non-linear difference equations in C. In particular, they showed in [1] that if the difference equation (1.1) f (z + 1) + f (z − 1) = R(z, f (z)) = a 0 (z) + a 1 (z)f (z) + · · · + a p (z)f (z) p b 0 (z) + b 1 (z)f (z) + · · · + b q (z)f (z) q admits a finite order meromorphic solution, then max(p, q) ≤ 2.…”

Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

“…Recently, there has been increasing interest in applying Nevanlinna theory to study meromorphic solutions of complex difference equations [20,21,48,58,67,80], and in particular, to detect integrability in discrete equations [1,49,50,111].…”

“…Equation (65) is a known integrable equation with continuum limits to Painlevé I and IV, and its Lax pair has been given in [36,39]. Equation (67) was found in connection with unitary matrix models of two-dimensional quantum gravity [101], and it was identified as the difference Painlevé II based on a continuum limit to the continuous Painlevé II equation. Equation (67) was also obtained as a similarity reduction of the discrete mKdV equation [94].…”

Wiman-Valiron theorem for q-differences

“…They obtained that ∆ k f (z) ∼ f (k) (z) holds outside an exceptional set. Ishizaki and Yanagihara [4] showed an analogue of Wiman-Valiron theory by thoughtfully rewriting power series of entire functions of order less than 1/2 into binomial series. Chiang and Feng [2] established a relationship between log f (z + q) − log f (z) and f ′ /f , further proved a difference analogue of Wiman-Valiron theory estimates for entire functions of order less than one, which can be used in the study of entire solutions of linear difference equations.…”