The properties of solutions for several types of Painlevé equations concerning fixed-points, zeros and poles

Abstract The purpose of this manuscript is to study some properties on meromorphic solutions for several types of q-difference equations. Some exponents of convergence of zeros, poles and fixed points related to meromorphic solutions for some q-difference equations are obtained. Our theorems are some extension and improvements to those results given by Qi, Peng, Chen, and Zhang.


Introduction and main results
Around 2006, Halburd-Korhonen [1] and Chiang-Feng [2] established independently some important fundamental results of Nevanlinna theory about the complex di erence and di erence operators. After their wonderful work, considerable attention has been paid in studying complex di erence equations, and a lot of important and interesting results (see [2][3][4]) focusing on complex di erence equations and di erence analogues of Nevanlinna theory were obtained. Halburd-Korhonen [1,5,6] studied the equation where R(z, f ) is rational in f and meromorphic in z, and they singled out the di erence Painlevé I equation (1.2) and the di erence Painlevé II equation Later, Ronkainen [7] in 2010 further discussed the equation where R(z, f ) is rational and irreducible in f and meromorphic in z. He pointed out that either f satis es the di erence Riccati equation or equation (1.4) can be transformed to one of the following equations where η(z), λ(z), υ(z) satisfy some conditions. The above four equations can be called as the di erence Painlevé III equations.
In what follows, we should assume that the readers are familiar with the fundamental theorems and the standard notations in the theory of Nevanlinna value distribution (see Hayman [8], Yang [9] and Yi-Yang [10]). Let f be a meromorphic function, we denote σ(f ), λ(f ) and λ( f ) to be the order, the exponent of convergence of zeros and the exponent of convergence of poles of f (z), respectively, and denote τ(f ) to be the exponent of convergence of xed points of f (z), which is de ned by In 2010, Chen-Shon [11] considered the di erence Painlevé I,II equation (1.2),(1.3) and obtained the following theorems. Theorem 1.1. (see [11,Theorem 4] In 2013 and 2014, Zhang-Yi [12], Zhang-Yang [13] studied the di erence Painlevé III equations with the constant coe cients, and obtained Theorem 1.3. (see [13]). If f (z) is a transcendental nite order meromorphic solution of (ii) f has at most one nonzero Borel exceptional value for σ(f ) > .
In 2007, Barnett, Halburd, Korhonen and Morgan [14] rst established the Logarithmic Derivative Lemma on complex q-di erence operators. Then by applying those fundamental results, many mathematicians have done a lot of work about the value distribution of complex q-di erence operators, solutions for complex q-di erence equations, by replacing the di erence f (z + c) with the q-di erence f (qz), q ∈ C \ { , } for the meromorphic function f (z) in some expression concerning complex di erence equations and complex di erence operators (see [15][16][17][18][19][20][21][22][23][24][25][26][27][28]). In 2015, Qi and Yang [29] considered the following equation which can be seen as q-di erence analogues of (1.2), and obtained the result as follows. Motivated by the idea from [29] and [13], our main aim of this aritcle is further to investigate some properties of meromorphic solutions for some q-di erence equations, which can be called as q-di erence Painlevé III equations. We obtain the following four results.
Theorem 1.6. Let q ∈ C\{ } and |q| ≠ , and let f (z) be a zero order transcendental meromorphic solution of the following equation where λ, µ are constants satisfying λµ ≠ . Then (i) for any η ∈ C − { , }, f (ηz) has in nitely many xed-points and τ(f (ηz)) = σ(f ), especially, f (q j z) has in nitely many xed-points and τ(f (q j z)) = σ(f ), where j is a positive integer; (ii) ∆q f , ∆q f f have in nitely many poles, and (iii) f (z) has in nitely many zeros and poles, and the Nevanlinna exceptional value of f (z) can only come from a set E = {z |z − z + λz − µ = }. Theorem 1.7. For q(≠ ) ∈ C and |q| ≠ , and let f (z) be a zero order transcendental meromorphic solution of the following equation Theorem 1.8. Let q ∈ C\{ } and |q| ≠ , and let f (z) be a zero order transcendental meromorphic solution of the following equation where λ(z) is a nonconstant polynomial. Then (i) for any η ∈ C\{ , }, f (ηz) has in nitely many xed-points and τ(f (ηz)) = σ(f ); (ii) f (z), ∆q f have in nitely many zeros and poles, and ∆q f f has in nitely many poles, and (iii) f (z) has no Nevanlinna exceptional value. Theorem 1.9. Let q ∈ C\{ } and |q| ≠ , and let f (z) be a zero order transcendental meromorphic solution of the following equation  Denote g(z) = f (ηz), then (2.1) can be represented as Thus, it follows In view of P (z, z) ≢ and by Theorem 2.5 in [14], it follows that m r, g(z) − z = S(r, g).
Thus, it follows from Theorem 1.1 in [14] that Hence, by combining with (2.10), we conclude that ∆q f has in nitely many poles and λ ∆q f = σ(f ).
Therefore, this completes the proof of Theorem 1.6 (ii).
(iii) By the process of the proof of Theorem 1.6 (ii), we have m(r, f ) = S(r, f ), this means N(r, f ) = T(r, f ) + S(r, f ), that is, ∞ is not a Nevanlinna exceptional value of f (z).
Besides, set Since µ ≠ , then it follows P (z, ) = µ ≠ . Thus, in view of Theorem 2.5 in [14], we have which implies that 0 is not a Nevanlinna exceptional value of f . Now, let β ∉ E, then it follow that From Theorem 2.5 in [14], it yields m r, f −β = S(r, f ), which implies that β is not a Nevanlinna exceptional value of f (z). Hence, the conclusion of Theorem 1.6 (iii) is true. Therefore, this completes the proof of Theorem 1.6.
. The proof of Theorem 1.7 By using the similar argument as in the proof of Theorem 1.6, it is easy to get the conclusions of Theorem 1.7.
Proofs of Theorems 1.8 and 1.9 . The proof of Theorem 1.8 Similar to the argument as in the proof of Theorem 1.6, we can prove that τ(f (ηz)) = σ(f ) and ∆q f , ∆q f f have in nitely many poles and λ ∆q f = λ ∆q f f . Now, we only need to prove that ∆q f has in nitely many zeros and λ(∆q f ) = σ(f ). We rewrite (1.8) as the form Besides, we can rewrite (1.8) again as the form Thus, in view of (3.2) and (3.3), and by Theorem 1.1 in [14], we can deduce that is, N(r, f ) = T(r, f ) + S(r, f ), which implies that f (z) has in nitely many zeros and λ(f ) = σ(f ).
, thus, substituting these into (1.8), it follows that Assume that z is a zero of f (z), in view of (3.5), then z must be a zero of ∆q f or ∆ q − f . Thus, in view of (3.4) and Theorem 1.1 in [14], it follows which implies that ∆q f has in nitely many zeros and λ(∆q f ) ≥ σ(f ). By combining with (2. Then for any a ∈ C\{ }, and since λ(z) is a nonconstant polynomial, we have P (z, a) = a (a − ) − aλ(z) ≢ . Thus, in view of Theorem 2.5 in [14], it yields that m r, f −a = S(r, f ), which means that δ(a, f ) = . Hence, it follows that f (z) has no Nevanlinna exceptional value. Therefore, this completes the proof of Theorem 1.8.
. The proof of Theorem 1.9 By using the similar argument as in the proof of Theorem 1.8, we can get the conclusions of Theorem 1.9 (i), (iii), ∆q f has in nitely many zeros and poles, and In view of (1.9), we have f (q z)f (z)f (qz) = h(qz). (3.6) Thus, from (1.9) and (3.6), it follows (3.7)