Meromorphic solutions of some linear functional equations

Abstract

Meromorphic solutions of non-linear differential equations of the form f n + P (z, f ) = h are investigated, where n ≥ 2 is an integer, h is a meromorphic function, and P (z, f ) is differential polynomial in f and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when h has the particular form h(z) = p 1 (z)e α1(z) + p 2 (z)e α2 (z) , where p 1 , p 2 are small functions of f and α 1 , α 2 are entire functions. In such a case the order of h is either a positive integer or equal to infinity. In this article it is assumed that h is a meromorphic solution of the linear differential equation h ′′ + r 1 (z)h ′ + r 0 (z)h = r 2 (z) with rational coefficients r 0 , r 1 , r 2 , and hence the order of h is a rational number. Recent results by Liao-Yang-Zhang (2013) and Liao (2015) follow as special cases of the main results.

FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS

“…The following result originates from [19,Theorem 3.5], which deals with the case of the usual order of growth.…”

“…, a n , a n+1 are entire and |q| > 1 at first. Following the proof of Theorem 3.5 in [19], we apply Lemma 3.3 to fix certain discs of radius |z j | −(ρ+ε) around the zeros z j of a n such that outside of these discs, |a n (z)| > exp(−(log r) ρ log (an)+ε ) for sufficiently large values of r. Let us fix T such that f has no poles with modulus |q| j T for any j ∈ N and these circles are outside of the discs of radius |z j | −(ρ+ε) . Then…”

“…Then γ q (z) is a meromorphic function of order zero with no zeros, having its poles at {q −k } ∞ k=0 , see [3]. Moreover, γ q (z) satisfies the q-difference equation (19) with A(z) = 1 − z. Hence, in fact, γ q (z) has logarithmic order 2.…”

“…Both of E q (z) and e q (z) satisfy (19) with A = 1 + z and A(z) = 1/(1 − z), respectively. Hence E q (z) and e q (z) have logarithmic order 2.…”

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

“…[3], [4], [5], [23], [36]), relatively little is known for the growth of meromorphic solutions to even the first order difference equation Given a finite order meromorphic coefficient Ψ(z), Whittaker [40, §6] explicitly constructed a meromorphic solution F (z) of order ≤ σ(Ψ)+1 to (1.4). On the other hand, let Π(z) be a periodic meromorphic function of period 1, then the product Π(z)F (z) again satisfies the equation (1.4).…”