Meromorphic solutions of difference Painlevé IV equations
© Zhang; licensee Springer. 2014
Received: 14 June 2014
Accepted: 17 September 2014
Published: 13 October 2014
In this paper, we investigate the family of difference Painlevé IV equation , where R is rational in w and meromorphic in z. If the equation assumes an admissible meromorphic solution of hyper-order , we fix the degree of , and we give some further discussions.
A meromorphic function is called a small function with respect to , if . The family of all meromorphic functions that are small with respect to f is denoted by . A meromorphic solution w of a difference equation is called admissible if all coefficients of the equation are in .
We conclude this section by the following expatiation on the coefficients. While we only consider meromorphic solutions of equations with meromorphic coefficients, we might encounter a situation where the coefficients have finitely-sheeted branchings, and we will use the algebroid version of Nevanlinna theory (see for instance ), which studies meromorphic functions on a finitely-sheeted Riemann surface. Whenever the coefficients have branchings, denotes the corresponding Nevanlinna characteristic function of a finite-sheeted algebroid function. Since all algebroid functions we need to consider are small functions with respect to the meromorphic solution of (1.2), the change in notation only affects the error term which needs to be redefined in terms of the algebroid characteristic. The ‘algebroid error term’ will still be denoted by and it remains small with respect to the meromorphic solution of (1.2), we can still denote it by .
2 Some lemmas
The difference analog of the logarithmic derivative lemma, which was obtained independently by Halburd and Korhonen  and by Chiang and Feng , plays a key role in the value distribution of difference [14–16]. The original version is valid for functions of finite order, and it was generalized to hold for meromorphic functions with hyper-order less than one recently.
Lemma 2.1 ()
for all r outside of a set of finite logarithmic measure.
We also need the following lemma to detect the hyper-order of a meromorphic function to be at least one.
as r tends to infinity outside of an exceptional set of finite logarithmic measure.
Then (2.1) follows from the last equation and (2.2). □
Without the order restriction, we have the following.
Lemma 2.5 ([, lemma 1])
for all .
If a meromorphic function f has a pole of the order n at , we denote this by . Similarly, an a-point of the order n is denoted by .
It could always happen that the coefficients in (1.2) have poles or zeros when w has a pole; whenever we meet these cases, we shall use the following result.
Lemma 2.6 ()
be the maximal order of zeros and poles of the functions at . Then for any there are at most points such that (or ) where .
The next result on the Nevanlinna characteristic is essential in the study of the family (1.2), for the proof, see, e.g., .
3 Main results
Definition 3.1 ()
P is said to be homogeneous if of each term in the sum (3.1) is nonzero and the same for all . The order of a zero of , as a function of at , is denoted by . Let be the maximum power of in , and let be the maximum power of in . Obviously, (). Denote .
A difference version of the Clunie lemma, developed by Halburd and Korhonen  and by Laine and Yang , works on such difference polynomials, say , having only one term of maximal degree in the sum (3.1). If is homogeneous, the above version does not work. We will establish a similar version of the Clunie lemma, for the other new type of version, see .
If , then . The coefficients of the highest degree of P and Q are identical;
If , then .
where and . For each , or for some .
Remark 1 In the present paper, we use similar notations to .
4 Proofs of theorems
which is , then from (3.2). □
Proof of Theorem 3.3 We restrict p and q duo to the reasoning by Grammaticos et al. , where and are polynomials in w with constant coefficients.
so . Thus . If , the degree of that was denoted by k would be 5 since , a contradiction. Hence, , , and .
If , since the degree of , , we have and the coefficients of the highest degree of P and Q are identical. □
for . From Lemma 2.6, given , there are at most points where , but where has a pole of order greater that or less than due to poles or zeros of . The combined effect of all such points can be included in the error term, and so we only consider the rest of the zeros of Q in what follows.
to be the longest possible list of points such that each is a zero of , , and each is a pole of w.
which implies that by Lemma 2.3. Therefore all except at most poles of w are in some sequence .
We will call the total number of zeros of in divided by the total number of poles of w (both counting multiplicities) the ratio of the sequence.
Both (4.7) and (4.8) hold .
Both (4.5) and (4.6) hold , (4.7) and (4.8) hold .
Relation (4.6) holds , (4.5), (4.7), and (4.8) hold .
Relation (4.5) holds , (4.6)-(4.8) hold .
where and .
Next, we will show that U is a small function with respect to w. From (4.3), we get . From the definition of U and the fact that all but at most poles of w are in sequences of the above form, it follows that if U has more than poles, then there are more than sequences where .
The same reasoning works for case (iv) as we exchange the roles of and .
In the case that , the proof is similar. The condition (i) is the only possibility. □
provided is not a solution of (3.3), i.e., does not satisfy . In the following, we may assume that has a large number of zeros, or (4.14) holds as desired.
for all except at most points and for either or both choices of ±.
We assume the condition (i) or (ii) is true, otherwise, w satisfies a Riccati difference equation. We discuss the following three cases:
Combining the above equation with (4.10) or (4.11), we have for or , which is a contradiction.
holds for or .
holds for or .
In this case, we have both (4.16) and (4.17).
where , , , and φ is a pole or a finite value but not the zero of . In fact, if φ is the zero of , it will be a starting point of another sequence.
If (4.20) holds for more than points, there are more than sequences such that , where . This is a contradiction. Hence, (4.20) holds at most points. Then almost all the zeros of are in the sequences (4.18) and (4.19).
However, noting that and , we have , which contradicts with Lemma 2.3.
The author would like to thank the referee for his or her valuable suggestions to the present paper. This research was supported by the NNSF of China Nos. 11201014, 11171013, 11126036 and the YWF-14-SXXY-008, YWF-ZY-302854, 301556 of Beihang University. This research was also supported by the youth talent program of Beijing No. 29201443.
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