Meromorphic Solutions of Linear Difference Equations with Polynomial Coefficients

We study the growth of the transcendental meromorphic solution f(z) of the linear difference equation:where q(z), p0(z), ..., pn-(z) (n ≥ 1) are polynomials such that p0(z)pn(z) ≢ 0, and obtain some necessary conditions guaranteeing that the order of f(z) satisfies σ(f) ≥ 1 using a difference analogue of the Wiman-Valiron theory. Moreover, we give the form of f(z) with two Borel exceptional values when two of p0(z), ..., pn(z) have the maximal degrees.

Milloux Inequality ofE-Valued Meromorphic Function

The main purpose of this paper is to establish the Milloux inequality ofE-valued meromorphic function from the complex planeℂto an infinite dimensional complex Banach spaceEwith a Schauder basis. As an application, we study the Borel exceptional values of anE-valued meromorphic function and those of its derivatives; results are obtained to extend some related results for meromorphic scalar-valued function of Singh, Gopalakrishna, and Bhoosnurmath.

Fixed Points of Difference Operator of Meromorphic Functions

Letfbe a transcendental meromorphic function of order less than one. The authors prove that the exact differenceΔf=fz+1-fzhas infinitely many fixed points, ifa∈ℂand∞are Borel exceptional values (or Nevanlinna deficiency values) off. These results extend the related results obtained by Chen and Shon.

Characteristic Functions and Borel Exceptional Values ofE-Valued Meromorphic Functions

The main purpose of this paper is to investigate the characteristic functions and Borel exceptional values ofE-valued meromorphic functions from theℂR={z:|z|<R}, 0<R≤+∞to an infinite-dimensional complex Banach spaceEwith a Schauder basis. Results obtained extend the relative results by Xuan, Wu and Yang, Bhoosnurmath, and Pujari.

Equi-Distribution of Values for the Third and the Fifth Painlevé Transcendents

We show equi-distribution properties of values for the third and the fifth Painlevé transcendents in a sectorial domain. For our purpose we define a characteristic function of sectorial domain type by employing value distribution theory in a half plane. Some special cases admit analogues of Borel exceptional values. Similar results are obtained for modified versions of these Painlevé transcendents, which are of infinite growth order.