# Some Properties of Solutions for Someq-Difference Equations Containing Painlevé Equation

Authors: Peng Jun Zhao; Hong Yan Xu
Journal: Journal of Function Spaces
 (2018)

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## Abstract

The existence and growth of meromorphic solutionsf(z) for some q-difference equations are studied, and some estimates for the exponent of convergence of poles of Δqf, Δq2f, Δqf/f, and Δq2f/f are also obtained. Our theorems are improvements and extensions of the previous results.


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## Body

## 1. Introduction and Main Results

In 1900, Painlevé [1] first studied the differential equations, which were called differential Painlevé equations later. Moreover, at the beginning of last century, differential Painlevé equations had been an important research subject in the field of the mathematics and physics. They occur in many physical situations, such as plasma physics, statistical mechanics, and nonlinear waves.In the 1990s, the discrete Painlevé equations had become important and interest research problems (see [2, 3]). For example, let a, b, and c be constants and n∈N+, it is usual that (1)yn+1+yn-1=an+byn+c,yn+1+yn-1=an+byn+cyn2are called the special discretization of discrete PI, and (2)yn+1+yn-1=an+byn+c1-yn2is called the special discretization of the discrete PII.Of late, with the development of Nevanlinna theory, Chiang and Feng [4] and Halburd and Korhonen [5] established independently those results about the difference analog of the lemma on the logarithmic derivative, and there has been an increasing interest in studying complex difference equations. And there were a number of papers (see [4, 6, 7]) concerning complex difference equations and difference analogs of Nevanlinna theory, by applying the results of Chiang and Feng [4] and Halburd and Korhonen [5]. For example, Halburd and Korhonen [5, 8, 9] used Nevanlinna theory to analyze the following equation:(3)wz+1+wz-1=Rz,w,where R(z,w) is rational in w and meromorphic in z, and we single out the difference of Painlevé I and II equations such as(4)wz+1+wz-1=az+bwz+c,(5)wz+1+wz-1=az+bwz+cwz2,(6)wz+1+wz-1=az+bwz+c1-wz2.In recent, Laine and Yang, Zhang and Korhonen, and Zheng and Chen further investigated the value distribution ofq-difference operator of meromorphic functions, by utilizing the analog of Logarithmic Derivative Lemma on q-difference operators given by Barnett et al. [10]. Moreover, during the last decades, considerable attention has been paid to q-difference operators, q-difference equations, by replacing the q-difference f(qz), q∈C∖{0,1} with f(z+c) of a meromorphic function in some complex difference equations and complex difference operators (see [11–26]).Throughout this paper, a term “meromorphic” will always mean meromorphic in the complex planeC. Hereinafter, we will use some basic results and the standard notations of Nevanlinna theory (see [27–29]). For a meromorphic function f, we use σ(f), λ(f), and λ(1/f) to denote the order, the exponent of convergence of zeros, and the exponent of convergence of poles of f(z), respectively, and let τ(f) be the exponent of convergence of fixed points of f(z), which is defined by(7)τf=lim⁡supr→+∞log⁡Nr,1/fz-zlog⁡r⁡.Besides, we useS(r,f) to denote any quantity satisfying S(r,f)=o(T(r,f)) for all r outside a possible exceptional set E of finite logarithmic measure (8)limr→∞⁡∫1,r∩Edtt<∞,and a meromorphic function a(z) is called a small function with respect to f if T(r,a(z))=S(r,f), and we use S(f) to denote the field of small functions relative to f(z).In 2010, Chen and Shon [30] considered the difference Painlevé I equation (4) and obtained the following theorem.Theorem 1 (see [30, Theorem 4]).
Leta, b, and c be constants, where a, b are not both equal to zero. Then
(i) ifa≠0, then (4) has no rational solution;
(ii) ifa=0, and b≠0, then (4) has a nonzero constant solution w(z)=A, where A satisfies 2A2-cA-b=0.
The other rational solutionw(z) satisfies w(z)=P(z)/Q(z)+A, where P(z) and Q(z) are relatively prime polynomials and satisfy deg⁡P<deg⁡Q.In 2015, the properties of solutions of a certain type of difference equation were further investigated by Li and Huang [31], and some results were obtained as follows.Theorem 2 (see [31, Theorem 3.1]).
Suppose that equation(9)wz+1+wz-1=Azwz+Cz,where A(z),C(z)∈S(w), admits a finite-order transcendental meromorphic solution w(z). Then
(i)λ(w)=λ(1/w)=σ(w);
(ii)w(z) has no Borel exceptional value;
(iii) ifA(z)≢2z2-zC(z), then the exponent of convergence of fixed points of w(z) satisfies τ(w)=σ(w).In the same year, Qi and Yang [32] discussed the following equation:(10)wqz+wzq=az+bwz+c,which can be seen as q-difference analogs of (4), and obtained some properties of the zeros of w(z)-A, where w(z) is a solution of (10) and A∈C∪{∞}.Inspired by the idea of Li and Huang [31] and Qi and Yang [32], our main purpose is further to investigate some properties of meromorphic solutions for some q-difference equations which are different from (10) to a certain extent, and the following theorems are obtained.Theorem 3.
Letq∈C-{0,1}, and A(z)(≢0),C(z)∈S(f). If(11)fqz+fzq=Azfz+Czadmits a zero-order transcendental meromorphic solution f(z), then
(i)f(z) has infinitely many poles and zeros, Δqf(z) also has infinitely many poles, and(12)λf=λ1f=λ1Δqf,and further, if C(z)≢0, then each of Δq2f, Δqf/f, Δq2f/f has infinitely many poles, and(13)λ1Δq2f=λ1Δqf/f=λ1Δq2f/f,
(ii) ifA(z)≢(q+1/q)z2-zC(z), then f has infinitely many fixed points and the exponent of convergence of fixed points of f satisfies τ(f)=σ(f).Theorem 4.
Letq∈C-{0,1}, and A(z),C(z)(≢0)∈S(f), and assume that f(z) is a zero-order transcendental meromorphic solution of equation(14)fqz+fzq=Azfz+Cz1-fz2.Then
(i)f has infinitely many poles and zeros, Δqf also has infinitely many poles, and(15)λf=λ1f=λ1Δqf,and further, if A(z)≢0, then each of Δq2f,Δqf/f,Δq2f/f has infinitely many poles, and(16)λ1Δq2f=λ1Δqf/f=λ1Δq2f/f,
(ii) if(q+1/q)(z3-z)+zA(z)≢-C(z), then f has infinitely many fixed points and the exponent of convergence of fixed points of f satisfies τ(f)=σ(f).Theorem 5.
Letq∈C-{0,1}, and A(z)(≢0),C(z)∈S(f) be identically vanishing simultaneously. And f(z) is a zero-order transcendental meromorphic solution of equation(17)fqz+fzq=Azfz+Czfz2.Then
(i)f(z) has infinitely many poles and zeros, Δqf(z) also has infinitely many poles, and(18)λf=λ1f=λ1Δqf,and further, if C(z)≢0, then each of Δq2f,Δqf/f,Δq2f/f has infinitely many poles, and(19)λ1Δq2f=λ1Δqf/f=λ1Δq2f/f,
(ii) if(q+1/q)z3-zA(z)≢C(z), then f has infinitely many fixed points and the exponent of convergence of fixed points of f satisfies τ(f)=σ(f).


## 2. Some Lemmas

Let the logarithmic density of a setF be defined by (20)lim⁡sup⁡r→∞1log⁡r⁡∫1,r∩F1tdt.Definition 6 (see [10]).
Forq∈C∖{0,1}, if a polynomial in f(z) includes finitely many of its q-shifts f(qz),…,f(qnz) with meromorphic coefficients in the sense that their Nevanlinna characteristic functions are o(T(r,f)) on a set F of logarithmic density 1, then it can be called a q-difference polynomial of f.Lemma 7 (see [13, Theorem 2.5]).
Letf be a transcendental meromorphic solution of order zero of a q-difference equation of the form (21)Uqz,fPqz,f=Qqz,f,where Uq(z,f), Pq(z,f), and Qq(z,f) are q-difference polynomials such that the total degree deg Uq(z,f)=n in f(z) and its q-shifts, whereas deg⁡Qq(z,f)≤n. Moreover, we assume that Uq(z,f) contains just one term of maximal total degree in f(z) and its q-shifts. Then(22)mr,Pqz,f=oTr,f,on a set of logarithmic density 1.Lemma 8 (see [10, Theorem 2.2]).
Letf be a nonconstant zero-order meromorphic solution of Pq(z,f)=0, where Pq(z,f) is a q-difference polynomial in f(z). If Pq(z,a)≢0 for slowly moving target α(z), then (23)mr,1f-α=oTr,f,on a set of logarithmic density 1.Remark 9 (see [10]).
Letα and f be meromorphic functions of zero-order such that T(r,α)=o(T(r,f)) on a set of logarithmic density 1. Then α is called a slowly moving target or a small function with respect to f.Lemma 10 (see [22, Theorems 1.1 and 1.3]).
Letf(z) be a nonconstant zero-order meromorphic function and q∈C∖{0}. Then (24)Tr,fqz=1+o1Tr,fz,Nr,fqz=1+o1Nr,fz,on a set of lower logarithmic density 1.Lemma 11 (Valiron-Mohon’ko, see [33]).
Letf(z) be a meromorphic function. Then for all irreducible rational functions in f,(25)Rz,fz=∑i=0maizfzi∑j=0nbjzfzj,with meromorphic coefficients ai(z), bj(z), the characteristic function of R(z,f(z)) satisfies (26)Tr,Rz,fz=dTr,f+OΨr,where d=max⁡{m,n} and Ψr=maxi,j{T(r,ai),T(r,bj)}.Lemma 12 (see [10, Theorem 1.1]).
Letf(z) be a nonconstant zero-order meromorphic function and q∈C∖{0}. Then(27)mr,fqzfz=Sr,fon a set of logarithmic density 1.


## 3. The Proof of Theorem3

We first assume thatf(z) is of zero-order and a transcendental meromorphic solution of (11).(i) In view of (11), it follows that(28)fzfqz+fzq=Czfz+Az.Hence, we conclude from (28) and Lemma 7 that(29)mr,fqz+fzq=Sr,f,on a set F of logarithmic density 1. Since f(z) is of zero-order, then we can deduce by Lemma 10 that(30)Nr,fqz+fzq≤Nr,fqz+Nr,fzq=21+o1Nr,f,on a set of lower logarithmic density 1.SinceA(z)≢0 and A(z),C(z)∈S(f), by applying Lemma 11 for (11), we obtain(31)Tr,fqz+fzq=Tr,f+Sr,f.Thus, it follows from (29) to (31) that(32)Tr,f≤21+o1Nr,f+Sr,f,on a set F of logarithmic density 1. Hence, it means that f(z) has infinitely many poles and(33)λ1f≥σf.On the other hand, it yields from (11) that(34)P1z,fz=fzfqz+fzq-Czfz-Az=0.Since A(z)≢0, then P1(z,0)=-A(z)≢0. Thus, by Lemma 8, it follows that(35)mr,1f=Sr,f,on a set F of logarithmic density 1. Hence,(36)Nr,1f=Tr,f+Sr,f,on a set F of logarithmic density 1. Therefore, f(z) has infinitely many zeros and λ(f)=σ(f).Next, we will prove thatλ1/Δqf≥λ1/f. Let z=qw, then (11) can be rewritten as the form(37)fq2w+fw=Aqwfqw+Cqw.Then it follows from (37) that(38)fqwfq2w+fw=Aqw+Cqwfqw.Moreover, it yields f(qw)=Δqf(w)+f(w) and f(q2w)=Δqf(qw)+Δqf(w)+f(w), where Δqf(w)=f(qw)-f(w). Substituting them into (38), we conclude(39)Δqfw+fwΔqfqw+Δqfw+2fw=Aqw+CqwΔqfw+fw,that is,(40)-2fw2=Δqfqw+3Δqfw-Cqwfw-Aqw+Δqfqw+Δqfw-CqwΔqfw.From the conditions of Theorem 3, and by Lemma 10, we conclude that f(w) is of zero order. Thus, we conclude that Δqf(w), Δqf(qw) are of zero order and A(qw),C(qw)∈S(f), and Δqf(qw)=Δq2f(w)+Δqf(w), where Δq2fw≔Δq(Δqf(w)). Since Δqf(w) is of zero order, then(41)Nr,Δq2fw≤2Nr,Δqfw+Sr,f,on a set F of logarithmic density 1. Thus, we can deduce from (41) that(42)Nr,Δqfqw≤3Nr,Δqfw+Sr,f,on a set F of logarithmic density 1. Moreover, it follows from (40) and (42) that(43)
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,on a set F of logarithmic density 1; that is,(44)Nr,fw≤9Nr,Δqfw+Sr,f,on a set F of logarithmic density 1. Then, it means that Δqf(w) has infinitely many poles and(45)λ1Δqfw≥λ1fw.Since f(w) is of zero order, then we conclude from Lemma 10 that (46)λ1Δqfw=λ1Δqfz/q=λ1Δqfz,λ1fw=λ1fz/q=λ1fz.Hence, it yields(47)λ1Δqf≥λ1f.Thus, by (33) and (47) we conclude that(48)λ1Δqf≥λ1f≥σf.T(r,Δqf)≤2T(r,f)+S(r,f) on a set F of logarithmic density 1: that is, σ(f)≥σ(Δqf). Thus, combining this and (48), we conclude that Δqf(z) has infinitely many poles and (49)λf=λ1f=λ1Δqf.Next, we prove that(50)λ1Δq2f=λ1Δqf/f=λ1Δq2f/f.At first, it can be seen thatA(z)+C(z)f(z) and f(z)2 are mutually prime polynomials in f(z), where A(z),C(z)∈S(f). In fact, taking u(z,f)=A(z)-C(z)f(z) and v(z,f)=C(z)2, it yields u(z,f)(A(z)+C(z)f(z))+v(z,f)f(z)2=A(z)2. Thus, in view of (11) and Lemma 11, we deduce (51)2Tr,f=Tr,Az+Czfzfz2+Sr,f=Tr,fqz+fz/qfz+Sr,f≤2Tr,fqzfz+Sr,f=2Tr,Δqff+Sr,f,that is, (52)Tr,f≤Tr,Δqff+Sr,f.Hence, by Lemma 12, it yields(53)Nr,Δqff=Tr,Δqff-mr,Δqff≥Tr,f+Sr,f,Nr,Δqff≤Tr,Δqff+Sr,f≤3Tr,f+Sr,f.Besides, in view of (11), it follows that(54)Δq2fzq=Δqfz-Δqfzq=-2fz2+Czfz+Azfz,and by Lemmas 10 and 11, it yields(55)2Tr,fz=Tr,-2fz2+Czfz+Azfz+Sr,f=Tr,Δ2fzq+Sr,f=Tr,Δq2fz+Sr,f.Thus, it follows that(56)3Tr,f+Sr,f≥Nr,Δq2f=Tr,Δq2f-mr,Δq2f≥2Tr,f-Tr,f+Sr,f=Tr,f+Sr,f.Finally, similar to the above argument, we have(57)2Tr,f+Sr,f≥Nr,Δq2ff=Tr,Δq2ff-mr,Δq2ff≥Tr,Δq2f-Tr,f+Sr,f=Tr,f+Sr,f.Thus, we can conclude from (53), (56), and (57) that each of Δq2f, Δqf/f, Δq2f/f has infinitely many poles and(58)λ1Δq2f=λ1Δqf/f=λ1Δq2f/f.(ii) Setg(z)=f(z)-z. Then g(z) is a zero-order transcendental meromorphic function with σ(g)=σ(f) and τ(f)=λ(g). Substituting f(z)=g(z)+z into (11), we have(59)P2z,gz=gz+zgqz+gzq+q+1qz-Czgz+q+1qz2-zCz-Az=0.Since P2(z,0)=(q+1/q)z2-zC(z)-A(z)≢0, then by Lemma 8, it yields(60)mr,1g=Sr,fon a set F of logarithmic density 1. By using the same argument as in the proof of Theorem 3(i), we conclude (61)Nr,1g=Tr,f+Sr,fon a set F of logarithmic density 1, which implies f has infinitely many fixed points and τ(f)=λ(g)=σ(f).Therefore, this completes the proof of Theorem3.


## 4. The Proof of Theorem4

We assume thatf(z) is a zero-order transcendental meromorphic solution of (14).(i) In view of (14), it follows that(62)Pqz,f≔fz2fqz+fzq+Azfz+Cz-fqz-fzq=0.Since C(z)≢0, then it follows that(63)Pqz,0≡Cz≢0.Thus, we can conclude by Lemma 8 that (64)mr,1f=Sr,f,on a set F of logarithmic density 1. Hence,(65)Nr,1f=Tr,f+Sr,f,on a set F of logarithmic density 1. Thus, it means that f(z) has infinitely many zeros and λ(f)=σ(f).On the other hand, it follows from (14) that(66)fz2fqz+fzq=fqz+fzq-Azfz-Cz.Then in view of (66) and Lemma 7, we have(67)mr,fqz+fzq=Sr,f.And by applying Lemma 11 for (14), it yields(68)Tr,fqz+fzq=2Tr,f+Sr,f.Then, we conclude by Lemma 10 that(69)Nr,fqz+fzq≤Nr,fqz+Nr,fzq=21+o1Nr,fon a set of lower logarithmic density 1. Thus, combining (67) and (68), it follows that(70)Tr,f≤21+o1Nr,f+Sr,f.Hence, f(z) has infinitely many poles and(71)σf≤λ1f.Next, we will prove thatλ1/Δqf≥λ1/f. Let z=qw; then (14) can be rewritten as the following form:(72)fq2w+fw=Aqwfqw+Cqw1-fqw2.Then from (72) it follows that(73)fqw2fq2w+fw=fq2w+fw-Aqwfqw-Cqw,and f(qw)=Δqf(w)+f(w) and f(q2w)=Δqf(qw)+Δqf(w)+f(w). Substituting them into (73), we have(74)Δqfw+fw2Δqfqw+Δqfw+2fw=Δqfqw+Δqfw+2fw-AqwΔqfw+fw-Cqw.Thus, it follows that(75)-2fw3=Pwfw+QwΔqfw+Rw,where (76)Pw=Δqfqw+5Δqfwfw+4Δqfw2+2ΔqfwΔqfqw+Aqw-2,Qw=ΔqfqwΔqfw+Δqfw2+Aqw-1,Rw=Cqw-Δqfqw.Hence, it follows by Lemma 10 and from (75) that(77)3Nr,fw=Nr,Pwfw+QwΔqfw+Rw≤2Nr,fw+19Nr,Δqfw+Olog⁡r+Sr,f,that is,(78)Nr,fw≤19Nr,Δqfw+Sr,f.Therefore, it follows from (78) that Δqf has infinitely many poles and(79)λ1Δqf≥λ1f.Since f(w) is of zero-order, by using the same argument as in the proof of Theorem 3, we conclude that Δqf(z) has infinitely many poles and(80)λ1f=λ1Δqf=σΔqf=σf.Here, we will prove that(81)λ1Δq2f=λ1Δqf/f=λ1Δq2f/f.Similarly, letA(z),C(z)∈S(f), u(z,f)=A(z)2f(z)2-A(z)C(z)f(z)+C(z)2-A(z)2, and v(z,f)=A(z)3; then u(z,f)(A(z)f(z)+C(z))+v(z,f)f(z)(1-f(z)2)=C(z)(C(z)2-A(z)2). Hence, it can be seen that A(z)f(z)+C(z) and f(z)(1-f(z)2) are mutually prime polynomials in f(z). Thus, from (14) and by Lemma 11, it follows that (82)3Tr,f=Tr,Az+Czfzfz1-fz2+Sr,f=Tr,fqz+fz/qfz+Sr,f≤2Tr,fqzfz+Sr,f=2Tr,Δqff+Sr,f,that is, (83)32Tr,f≤Tr,Δqff+Sr,f.Hence, in view of Lemmas 8 and 12, it yields(84)Nr,Δqfzfz=Tr,Δqfzfz-mr,Δqfzfz≥32Tr,fz+Sr,f,Nr,Δqff≤Tr,Δqff+Sr,f≤3Tr,f+Sr,f.Besides, in view of (14), we have (85)Δq2fzq=Δqfz-Δqfzq=2fz3+Az-2fz+Cz1-fz2,and then by Lemma 11, it follows that(86)3Tr,fz=Tr,2fz3+Az-2fz+Cz1-fz2+Sr,f=Tr,Δq2fzq+Sr,f=Tr,Δq2fz+Sr,f.Thus, by Lemma 10 and the above equalities, it yields(87)3Tr,f+Sr,f≥Nr,Δq2f=Tr,Δq2fz-mr,Δq2f≥3Tr,f-Tr,f+Sr,f=2Tr,f+Sr,f.Finally, similar to the above argument, it follows that(88)3Tr,f+Sr,f≥Nr,Δq2ff=Tr,Δq2ff-mr,Δq2ff≥Tr,Δq2f-Tr,f+Sr,f=2Tr,f+Sr,f.Thus, in view of (84), (87), and (88), we can deduce that each of Δq2f, Δqf/f, Δq2f/f has infinitely many poles and(89)λ1Δq2f=λ1Δqf/f=λ1Δq2f/f.(ii) Setg(z)=f(z)-z. Thus, g(z) is a zero-order transcendental meromorphic function with σ(g)=σ(f) and τ(f)=λ(g). Substituting f(z)=g(z)+z into (14), it follows that(90)P2z,gz=gz+z2gqz+gzq+gzgz+2zq+1qz-gqz+gzq+Azgz+q+1qz3-z+zAz+Cz.Since P2(z,0)=(q+1/q)(z3-z)+zA(z)+C(z)≢0, then in view of Lemma 8, it follows that(91)mr,1g=Sr,fon a set F of logarithmic density 1. By using the same argument as in the proof of Theorem 3(i), it follows that(92)Nr,1g=Tr,f+Sr,fon a set F of logarithmic density 1, which implies that f(z) has infinitely many fixed points and τ(f)=λ(g)=σ(f).Therefore, this completes the proof of Theorem4.


## 5. The Proof of Theorem5

We first assume thatf(z) is a zero-order transcendental meromorphic solution of (17).(i) In view of (17), it follows that(93)fz2fqz+fzq=Azfz+Cz.Thus, we conclude from (93) and Lemma 7 that(94)mr,fqz+fzq=Sr,f.And by Lemma 10 it follows that(95)Nr,fqz+fzq≤Nr,fqz+Nr,fzq=21+o1Nr,f,on a set of lower logarithmic density 1.IfC(z)≠0, then by applying Lemma 11 for (17), it follows that(96)Tr,fqz+fzq=2Tr,f+Sr,f.Hence, from (94) to (96), it yields(97)Tr,f≤1+o1Nr,f+Sr,f.IfA(z)≠0 and C(z)≡0, by using the same argument as in the proof of Theorem 3(i), it follows that(98)Tr,f≤21+o1Nr,f+Sr,f.Therefore, from (97) and (98), it means that f(z) has infinitely many poles and(99)σf≤λ1f.Now, we will prove thatλ1/Δqf≥λ1/f as follows.Setz=qw, by using the same argument as in Theorem 3(i), it yields(100)-2fw3=Pwfw+QwΔqfw+Rw,where(101)Pw=Δqfqw+5Δqfwfw+4Δqfw2+2ΔqfwΔqfqw-Aqw,Qw=Δqfqw+ΔqfwΔqfw-Aqw,Rw=-Cqw.Hence, we conclude by Lemma 10 and from (100) that(102)3Nr,fw=Nr,Pwfw+QwΔqfw+Rw≤2Nr,fw+15Nr,Δqfw+Olog⁡r+Sr,f,that is,(103)Nr,fw≤15Nr,Δqfw+Sr,f.Thus, it follows from (103) that(104)λ1Δqfw≥λ1fw.By using the same argument as in the proof of Theorem 3(i), we can conclude that Δqf has infinitely many poles, and (105)λ1f=λ1Δqf=σΔqf=σf,λ1Δq2f=λ1Δqf/f=λ1Δq2f/f.(ii) Letg(z)=f(z)-z. Thus, g(z) is a zero-order transcendental meromorphic function with σ(g)=σ(f) and τ(f)=λ(g). Substituting f(z)=g(z)+z into (17), it follows that(106)P2z,gz=gz+z2gqz+gzq+gzgz+2zq+1qz-Azgqz+q+1qz3-zAz-Cz.Since P2(z,0)=(q+1/q)z3-zA(z)-C(z)≢0, then it yields by Lemma 8 that(107)mr,1g=Sr,fon a set F of logarithmic density 1. By using the same argument as in the proof of Theorem 3(i), we have (108)Nr,1g=Tr,f+Sr,fon a set F of logarithmic density 1, which implies that f(z) has infinitely many fixed points and τ(f)=λ(g)=σ(f).Therefore, this completes the proof of Theorem5.



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