Results on Solutions for Several q-Painlevé Difference Equations concerning Rational Solutions, Zeros, and Poles

Bu Sheng Li, Rui Ying, Xiu Min Zheng , and Hong Yan Xu 4 Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen 333403, Jiangxi, China Basic Department, Shangrao Preschool Education College, Shangrao 334001, Jiangxi, China Institute of Mathematics and Information Science, Jiangxi Normal University, Nanchan 330022, Jiangxi, China School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, Jiangxi, China


Introduction and Main Results
All the time, Painlevé equations have attracted much interest due to the reduction of solution equations, which are solvable by inverse scattering transformations, and they often occur in many physical situations: plasma physics, statistical mechanics, and nonlinear waves. e study of Painlevé equations has spanned more than one hundred years (see [1][2][3]).
Around 2006, Halburd and Korhonen [4,5] and Ronkainen [6] used Nevanlinna theory to discuss the following equations: where R(z, f) is rational in f and meromorphic in z, respectively, and they singled out the following difference equations: where η(z), λ(z), and υ(z) satisfy some conditions. In these equations, equation (2) is called as the difference Painlevé I equation, equation (3) is called as the difference Painlevé II equation, and the last four equations are called as the difference Painlevé III equations. In the last decade or so, there were a lot of papers focusing on the properties of solutions for difference Painlevé I-IV equations (see [7][8][9][10][11]). For example, Chen and Shon [12] in 2010 considered the difference Painlevé I equation (3) and obtained the following theorem.
Theorem 1 (see [12], eorem 4). Let a, b, c be constants, where a, b are not both equal to zero. en, the following holds: (i) If a ≠ 0, then (3) has no rational solution.
In 2013 and 2018, Zhang and Yi [11] and Du et al. [13] studied the difference Painlevé III equations with the constant coefficients and obtained the result as follows.
Theorem 2 (see [11,13]). If f is a transcendental finite-order meromorphic solution of (8) where λ and μ are constants, then the following holds: Ramani et al. [14] in 2003 investigated the existence of transcendental solution of equation which is called as difference Painlevé IV equations and obtained the result as follows.
Theorem 3 (see [14]). If the second-order difference equation (9) admits a nonrational meromorphic solution of finite order, then deg z P ≤ 4 and deg z Q ≤ 2.
Of late, many mathematicians paid considerable attention to the value distribution of solutions for complex q-difference equations, which are formed by replacing the q-difference f(qz), q ∈ C/ 0, 1 { } with f(z + c) of meromorphic function in some expressions concerning complex difference equations, by utilizing the logarithmic derivative lemma on q-difference operators given by Barnett et al. [15] in 2007 (see [16][17][18][19][20][21][22][23][24][25][26]). For example, Qi and Yang [27] considered the following equation: which can be seen as q-difference analogues of (2) and obtained the result as follows.
Theorem 4 (see [27], eorem 1). Let f(z) be a transcendental meromorphic solution with zero order of equation (10) and a, b, c be three constants such that a, b cannot vanish simultaneously. en, the following holds: (i) f(z) has infinitely many poles. In 2018, Liu and Zhang [28] further investigated the following equation: and obtained the result as follows.
Theorem 5 (see [28], eorem 1). Let Y(z) be a transcendental meromorphic solution with zero order of (11) and ξ, o, ] be three constants such that ξ, o cannot vanish simultaneously. en, the following holds: Motivated by the idea [27,28], a natural question is what is the result if we give q-difference analogues of (9). For this question, our main aim of this article is further to investigate some properties of meromorphic solutions for some q-Painlevé difference IV equations. It seems that this topic has never been treated before.
In what follows, it should be assumed that the readers are familiar with the fundamental results and the standard notations in the theory of Nevanlinna value distribution (see Hayman [29], Yang [30], and Yi and Yang [31]). Let f be a meromorphic function, and we denote σ(f), λ(f), and λ(1/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 fixed points of f(z), which is defined by 2

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In addition, we use S(r, f) denotes any quantity satisfying S(r, f) � o(T(r, f)) for all r on a set F of logarithmic density 1, and the logarithmic density of a set F is defined by Now, our main results are listed as follows.
be an irreducible rational function, and let q( ≠ 0) ∈ C and |q| ≠ 1, and where (i) Suppose that a ≥ b and a − b are even numbers or zero. If equation (14) has an irreducible rational then the following holds: (ii) Suppose that b ≥ a and b − a are even numbers or zero. If equation (14) has an irreducible rational solution (iii) If |a − b| is an odd number, then equation (14) has no rational solution.

Theorem 8.
For q( ≠ 0) ∈ C and |q| ≠ 1, and let f(z) be a transcendental meromorphic solution with zero order of equation where

d(z) is a nonconstant rational function satisfying that d(qz)/d(z)
is not a constant. en, the following holds: (i) Both f and Δ q f/f have no Nevanlinna exceptional value. (ii) Δ q f has infinitely many poles and zeros, and
(i) Suppose that a > b and a − b are even numbers. en, in view of (14) and (19), it yields However, A(z)/B(z) ⟶ ∞ as z ⟶ ∞; thus, from (20), we can get a contradiction easily. If μ � ], then let z ⟶ ∞, and it leads to Journal of Mathematics where α is a nonzero constant. us, let z ⟶ +∞; in view of (20), we also get a contradiction. So, it follows μ > ]. us, assume that a s ≠ 0, where β( ≠ 0) is a constant, and it follows now in view of (20) that as z ⟶ ∞. Since |q| ≠ 1, then q s + 2 + q − s ≠ 0. Hence, it follows from (24) that Next, assume that a � b. As z ⟶ ∞, it follows where β( ≠ 0) is a constant. If μ < ], then by using the same argument as above, we get a contradiction. If μ ≥ ], then we assume that a s ≠ 0, (s ≥ 1). By using the same argument as above, we conclude as z ⟶ ∞. us, if μ > ], then in view of (27), we can get a contradiction; if μ � ], then we have (ii) Suppose that b > a and b − a are even numbers. en, in view of (14) and (19), we get (20). If μ > ], then for z ⟶ ∞, it leads to However, A(z)/B(z) ⟶ 0 as z ⟶ ∞; thus, from (20), we can get a contradiction easily. If μ � ], then let z ⟶ ∞, it follows where α is a nonzero constant. us, let z ⟶ +∞; in view of (20), we also get a contradiction. us, μ < ]. We rewrite (14) as the following form: Denote where a ≥ 1, b ≥ 0, and μ ≥ 0, ] ≥ 1 are all nonnegative integers. us, in view of (31) and (32), we can deduce Since |q| ≠ 1, then q μ− ] + 2 + q ]− μ ≠ 0. us, by combining with this and (33), we have and Assume that f(z) � μ(z)/](z) is a rational solution of (14). In view of the conclusion of eorem 6 (i), it follows μ − ] � (a − b)/2. is means a contradiction with the assumption that a − b is an odd number. us, (14) has no rational solution.
If a < b, then |a − b| � b − a is an odd number. Similar to the above argument, we also conclude that (14) has no rational solution. 4 Journal of Mathematics erefore, this completes the proof of eorem 6.

Proof of Theorem 7
We first introduce some notations and some basic results about Nevanlinna theory, which can be used in Section 3 and Section 4. Let f be a meromorphic function in C, the Nevanlinna characteristic T(r, f), which encodes information about the distribution of values of f on the disk |z| ≤ r, is defined by e proximity function m(r, f) is defined by where log + x � max 0, log x and where n(r, f) is the number of poles of f in the circle |z| � r, counted according to multiplicities.
If δ(a, f) > 0, then the complex number a is called the Nevanlinna exceptional value. And the order σ(f), the exponent of convergence of zeros λ(f), and the exponent of convergence of poles λ(1/f) of f(z) are defined by Besides, we also use some properties of T(r, f), m(r, f), and N(r, f) such as where f j (z)(j � 1, 2, . . . , p) are p meromorphic functions and a ∈ C and require some lemmas as follows.
Lemma 1 (see [15], eorem 2). Let f be a nonconstant zero-order meromorphic solution of where S(r, f) denotes any quantity satisfying S(r, f) � o(T(r, f)) for all r on a set F of logarithmic density 1.

Remark 1.
For q ∈ C/ 0, 1 { }, a polynomial in f(z) and finitely many of its q-shifts f(qz), . . . , f(q n z) with meromorphic coefficients in the sense that their Nevanlinna characteristic functions are o (T(r, f)) on a set F of logarithmic density 1 and can be called as a q-difference polynomial of f. Lemma 2 (see [24], eorems 1 and 3). Let f(z) be a nonconstant zero-order meromorphic function and q ∈ C/ 0 { }. en, on a set of lower logarithmic density 1.

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e proof of eorem 7: (i) suppose that f(z) is a transcendental meromorphic solution of equation (17), then in view of (17), let For any given constant d ∈ C/ 0 { } and with a view of a ≠ 4, it follows In view of P 1 (z, d) ≢ 0 and by Lemma 1, we conclude that m(r, which implies δ(d, f) � 0. us, f(z) has no nonzero finite Nevanlinna exceptional value.
Since f(z) is of zero order and Δ q f � f(qz) − f(z), then by Lemma 2, it follows T(r, Δ q f) ≤ 2T(r, f) + S(r, f), which means that Δ q f if of zero order. In view of (17), it follows With (17) subtraction, it leads to From (17), we see that f(qz) + f(z) ≡ 0. Otherwise, it leads to f(z) ≡ 0, a contradiction. us, the above equality means that is, Denote For any given constant d ∈ C/ 0 { }, then from (50), we have P 2 (z, d) � (4 − a)d. Hence, with a view of a ≠ 4, it follows P 2 (z, d) ≡ 0.

Proof of Theorem 8
Lemma 3 (see [15], eorem 1). Let f(z) be a nonconstant zero-order meromorphic function and q ∈ C/ 0 { }. en, Lemma 4 (see [17], eorem 2.5). Let f be a transcendental meromorphic solution of order zero of a q-difference equation of the form where U q (z, f), P q (z, f), and Q q (z, f) are q-difference polynomials such that the total degree deg U q (z, f) � n in f(z) and its q-shifts, whereas deg Q q (z, f) ≤ n. Moreover, we assume that U q (z, f) contains just one term of maximal total degree in f(z) and its q-shifts. en, Proof of eorem 8:. (i) suppose that f(z) is a transcendental meromorphic solution of equation (18). We firstly prove that Δ q f/f has no Nevanlinna exceptional value. Equation (18) can be rewritten as Set g 3 (z) � f(qz)/f(z). In view of (61) and Lemma 2, it follows And with a view of T(r, Δ q f/f) ≤ 2T(r, f) + S(r, f), we thus conclude that Δ q f/f is transcendental and of order zero, and d(z) is small with respect to g 3 (z).
Next, we prove that f(z) has no Nevanlinna exceptional value.
Firstly, in view of (61) and Lemma 3, we have which implies N r, 1 f � T(r, f) + S(r, f).