Nevanlinna Theory for Jackson Difference Operators and Entire Solutions of q-Difference Equations

This paper has two purposes. One is to establish a version of Nevanlinna theory based on the historic so-called Jackson difference operator D q f ( z ) = f ( q z ) − f ( z ) q z − z for meromorphic functions of zero order in the complex plane ℂ. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem, five-value theorem and Wiman-Valiron theorem in sense of Jackson q -difference operator, which keep consistent with the classical Nevanlinna theory. The other is, by using this theory, to investigate entire solutions of linear q -difference equations concerning Jackson difference operators. We will study the growth of entire solutions of linear Jackson q -difference equations D q k f ( z ) + A ( z ) f ( z ) = 0 with meromorphic coefficient A , where D q k is Jackson k -th order difference operator, and then to estimate the logarithmic order of some q -special functions. Further, we show that the growth of order of all admissible meromorphic solutions f of a general linear nonhomogeneous q -difference equations A n ( z ) D q n f ( z ) + ⋯ + A 1 ( z ) D q f ( z ) + A 0 ( z ) f ( z ) = F ( z ) should be positive if δ (0, f ) > 0.