Value distribution of q -differences of meromorphic functions in several complex variables
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VALUE DISTRIBUTION OF q-DIFFERENCES OF MEROMORPHIC FUNCTIONS IN SEVERAL COMPLEX VARIABLES T.-B. CAO1,† and R. J. KORHONEN2,∗,‡ 1 2
Department of Mathematics, Nanchang University, Jiangxi 330031, P. R. China e-mail: [email protected]
Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland e-mail: [email protected] (Received May 17, 2019; revised June 7, 2019; accepted June 22, 2020)
Abstract. In this paper, we study q-difference analogues of several central results in value distribution theory of several complex variables such as q-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura–Clunie theorem concerning partial q-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial q-difference equations.
1. Introduction At the beginning of the twentieth century, the foundations of the general theory of the linear difference equations in one independent variable was built by N¨orlund [51], Carmichael [15] and Pincherle [52], and of the linear q-difference equations by Jackson [38], Mason [46,47] and Adams [4]. Meanwhile, the general theory of difference equations in more than one independent variable, or of linear partial difference equations, was studied by Adams [1–3,5]. For this background of linear partial q-difference equations, we refer to a review article of Adams [6]. Since 1925, when R. Nevanlinna [49] established the theory, the value distribution theory of meromorphic functions of one complex variable has ∗ Corresponding
author. first author was supported by the National Natural Science Foundation of China (#11871260, #11461042), and the outstanding young talent assistance program of Jiangxi Province (#20171BCB23002) in China. ‡ The second author was supported in part by the Academy of Finland grant (#286877). Key words and phrases: logarithmic derivative lemma, second main theorem, partial q-difference equation, Picard type theorem, Tumura–Clunie theorem. Mathematics Subject Classification: primary 32H30, secondary 30D35, 39A14. † The
c 2020 Akad´ 0133-3852 emiai Kiad´ o, Budapest
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T.-B. CAO and R. J. KORHONEN
found numerous applications in the theory of complex differential equations. The most striking result, in the Nevanlinna theory of value distribution, is the second main theorem, which is an inequality relating two leading quantities in the value distribution theory, namely, the characteristic function, which measures the rate of growth of a function or a map, and the counting function, which tells the size of the preimages of points or sets. Later on, many forms of the second main theorem for holomorphic maps, as well as meromorphic maps, on various contexts were found. For example, in 1933, H. Cartan [16] extended Nevanlinna’s second main theorem for the case of holomorphic curves into complex projective spaces sharing hyperplanes in general positi
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