Growth of meromorphic solutions of delay differential equations

Transcription

1 Growth of meromorphic solutions of delay differential equations Rod Halburd and Risto Korhonen 2 Abstract Necessary conditions are obtained for certain types of rational delay differential equations to admit a non-rational meromorphic solution of hyper-order less than one. The equations obtained include delay Painlevé equations and equations solved by elliptic functions. Introduction There have been many studies of the discrete or difference Painlevé equations. One way in which difference Painlevé equations arise is in the study of difference equations admitting meromorphic solutions of slow growth in the sense of Nevanlinna. The idea that the existence of sufficiently many finite-order meromorphic solutions could be considered as a version of the Painlevé property for difference equations was first advocated in []. This is a very restrictive property, as demonstrated by the relatively short list of possible equations obtained in [3] of the form wz + + wz = Rz, wz, where R is rational in w with meromorphic coefficients in z, and w is assumed to have finite order but to grow faster than the coefficients. It was later shown in [4] that the same list is obtained by replacing the finite order assumption with the weaker assumption of hyper-order less than one. Some reductions of integrable differential-difference equations are known to yield delay differential equations with formal continuum limits to differential Painlevé equations. For example, Quispel, Capel and Sahadevan [8] obtained the equation wz [wz + wz ] + aw z = bwz,. where a and b are constants, as a symmetry reduction of the Kac-van Moerbeke equation. They showed that equation. has a formal continuum limit to the first Painlevé equation d 2 y dt 2 = 6y2 + t..2 Furthermore, they obtained an associated linear problem for equation. by extending the symmetry reduction to the Lax pair for the Kac-van Moerbeke equation. Painlevé-type delay differential equations were also considered in Grammaticos, Ramani and Moreira [2] from the point of view of a kind of singularity confinement. More recently, Viallet [0] has introduced a notion of algebraic entropy for such equations. We will assume that the reader is familiar with the standard notation and basic results of Nevanlinna theory see, e.g., [5]. Let wz be a meromorphic function. The hyper-order or the iterated order of wz is defined by ρ 2 w = lim sup r log log T r, w, log r Department of Mathematics, University College London, Gower Street, London WCE 6BT, UK. r.halburd@ucl.ac.uk 2 Department of Physics and Mathematics, University of Eastern Finland, P.O. Box, FI-800 Joensuu, Finland. risto.korhonen@uef.fi

2 where T r, w is the Nevanlinna characteristic function of w. Most of the present paper is devoted to a proof of the following. Theorem.. Let wz be a non-rational meromorphic solution of wz + wz + az w z wz = Rz, wz = P z, wz Qz, wz,.3 where az is rational, P z, w is a polynomial in w having rational coefficients in z, and Qz, w is a polynomial in wz with roots that are nonzero rational functions of z and not roots of P z, w. If the hyper-order of wz is less than one, then deg w P = deg w Q + 3 or deg w R..4 We have used the notation deg w P = deg w P z, w for the degree of P as a polynomial in w and deg w R = max{deg w P, deg w Q} for the degree of R as a rational function of w. If deg w Rz, w = 0 then equation.3 becomes wz + wz + az w z wz = bz,.5 where az and bz are rational. Note that if bz pπiaz, where p N, then wz = C exppπiz, C 0, is a one-parameter family of zero-free entire transcendental finiteorder solutions of.5 for any rational az. In the following theorem we will single out the equation. from the class.5 by introducing an additional assumption that the meromorphic solution has sufficiently many simple zeros. In value distribution theory the notation Sr, w usually means a quantity of magnitude ot r, w as r outside of an exceptional set of finite linear measure. In what follows we use a slightly modified definition with a larger exceptional set of finite logarithmic measure. We use the notation Nr, w to denote the integrated counting function of poles counting multiplicities and Nr, w to denote the integrated counting function of poles ignoring multiplicities. Theorem.2. Let wz be a non-rational meromorphic solution of equation.5, where az 0 and bz are rational. If the hyper-order of wz is less than one and for any ɛ > 0 N r, 3 w 4 + ɛ T r, w + Sr, w,.6 then the coefficients az and bz are both constants. Finally, we consider an equation outside the class.3. Theorem.3. Let wz be a non-rational meromorphic solution of wz + wz = azw z + bzwz wz 2 + cz,.7 where az 0, bz and cz are rational. If the hyper-order of wz is less than one and for any ɛ > 0 N r, 3 w 4 + ɛ T r, w + Sr, w,.8 then.7 has the form wz + wz = λ + µzw z + νλ + µνz wz wz 2,.9 where λ, µ and ν are constants. 2

3 When µ = ν = 0 and λ 0 then equation.9 has a multi-parameter family of elliptic function solutions: wz = α [ Ωz; g 2, g 3 Ω; g 2, g 3 ], where is the Weierstrass elliptic function, Ω, g 2 and g 3 are arbitrary provided that Ω; g 2, g 3 0 or and α 2 = λω/ Ω; g 2, g 3. Furthermore, when µ = 0, equation.9 has a formal continuum limit to the first Painlevé equation. Specifically, we take the limit ɛ 0 for fixed t = ɛz, where wz = ɛ 2 yt, λ = 2 + Oɛ and λν = 3 ɛ5 + Oɛ 6. Then equation.9 becomes d 3 y/dt 3 = 2y dy/dt +, which integrates to d 2 y/dt 2 = 6y 2 + t t 0, for some constant t 0. Replacing t with t + t 0 gives the first Painlevé equation.2. Finally, when µ = 0 and λν 0, equation.9 is a symmetry reduction of the known integrable differential-difference modified Korteweg-de Vries equation v t x, t = vx, t 2 vx +, t vx, t, in which vx, t = 2λνt /2 wz, where z = x 2ν log t. 2 Value distribution of slow growth solutions We begin by proving an important lemma, which relates the value distribution of meromorphic solutions of a large class of delay differential equations to the growth of these solutions. A differential difference polynomial in wz is defined by P z, w = l L b l zwz l 0,0 wz + c l,0 wz + c ν l ν,0 w z l 0, w µ z + c ν lν,µ, where c,..., c ν are distinct complex constants, L is a finite index set consisting of elements of the form l = l 0,0,..., l ν,µ and the coefficients b l z are rational functions of z for all l L. Lemma 2.. Let wz be a non-rational meromorphic solution of P z, w = 0 2. where P z, w is differential difference polynomial in wz with rational coefficients, and let a,..., a k be rational functions satisfying P z, a j 0 for all j {,..., k}. If there exists s > 0 and τ 0, such that k j= n r, then the hyper-order ρ 2 w of w is at least. kτ nr + s, w + O, 2.2 w a j Proof. We suppose against the conclusion that ρ 2 w < aiming to obtain a contradiction. We first show that the assumption P z, a j 0 implies that m r, = Sr, w. 2.3 w a j This fact is an extension of Mohon ko s theorem and its difference analogue see [4, Remark 5.3] for differential delay equations with meromorphic solutions of hyper-order less than one. 3

4 By substituting w = g + a j into 2. it follows that Qz, g + Rz = 0, 2.4 where Rz 0 is a rational function, and Qz, g = b l zg l z, g 2.5 l L is a differential difference polynomial in g such for all l in the finite index set L, G l z, g is a non-constant product of derivatives and shifts of gz. The coefficients b l in 2.5 are all rational. Now, letting E = {θ [0, 2π : gre iθ } and E 2 = [0, 2π \ E, we have m r, = m w a j r, g = log + θ E gre iθ dθ 2π. 2.6 Moreover, for all z = re iθ such that θ E, Qz, g g = b g l zgz l 0,0 gz + c l,0 gz + c ν l ν,0 g z l 0, g µ z + c ν lν,µ l L b l z gz + c,0 gz l gz + c ν ν,0 gz l g z 0, g gz l µ z + c ν gz l L since deg g G l for all l L with l = l 0,0,..., l ν,µ. Now, since log + gz Rz log+ gz + log+ Rz = log + Qz, g gz + log+ Rz by equation 2.4, it follows from 2.6 by defining c 0 = 0 that m r, log + Qz, g dθ w a j θ E gz + Olog r 2π ν µ l n,m m r, gm z + c n + Olog r gz n=0 m=0 ν µ l n,m m r, gm z + c n gz + c n n=0 m=0 + m r, gz + c n + Olog r. gz l ν,µ, 2.7 The claim that 2.3 holds follows by applying the lemma on the logarithmic derivative, its difference analogue [4, Theorem 5.] and [4, Lemma 8.3], to the right hand side of 2.7. To finish the proof, we observe that from the assumption 2.2 it follows that k N j= r, τ + εk Nr + s, w + Olog r 2.8 w a j where ε > 0 is chosen so that τ + ε <. The first main theorem of Nevanlinna theory now yields k kt r, w = m r, + N r, + Olog r. 2.9 w a j w a j j= 4

5 By combining 2.3, 2.8 and 2.9 it follows that kt r, w τ + εknr + s, w + Sr, w τ + εk T r + s, w + Sr, w. 2.0 An application of [4, Lemma 8.3] yields T r + s, w = T r, w + Sr, w, and so 2.0 becomes T r, w τ + εt r, w + Sr, w, which gives us the desired contradiction T r, w = Sr, w since τ + ε <. We conclude that ρ 2 w. 3 The proof of Theorem. Before proving Theorem. we first prove three lemmas related to equations of the form.3. The first bounds the degree of R. Lemma 3.. Let wz be a non-rational meromorphic solution of equation.3 where a and R are rational functions of one and two variables respectively. If the hyper-order of w is less than one then deg w R 4. Furthermore, if the hyper-order of w is less than one and deg w R = 4 then Nr, /w = T r, w + Sr, w. Proof. Taking the Nevanlinna characteristic function of both sides of.3 and applying an identity due to Valiron [9] and Mohon ko [7] see also [6, Theorem 2.2.5], we have T r, wz + wz + az w z = T r, Rz, wz wz = deg w RT r, wz + Olog r. Thus by using the lemma on the logarithmic derivative and its difference analogue [4, Theorem 5.], it follows that deg w RT r, wz T r, wz + wz + T r, w z + Olog r wz N r, wz + wz + mr, wz + Nr, wz + N r, + Sr, w. wz 3. On using [4, Lemma 8.3] to obtain N r, wz + wz N r, wz + + N r, wz 2Nr +, wz = 2Nr, wz + Sr, w, inequality 3. becomes deg w RT r, wz T r, wz + Nr, wz + Nr, wz + N r, + Sr, w. wz 3.2 Therefore deg w R 3T r, wz N which implies the conclusions of the lemma. 5 r, + Sr, w, 3.3 wz

6 Next we consider the case in which Rz, w is a polynomial in w. Lemma 3.2. Let w be a non-rational meromorphic solution of the equation wz + wz + az w z wz = P z, wz, 3.4 where az is rational in z and P z, w is a polynomial in w and rational in z. If the hyper-order of w is less than one then deg w P. Proof. Assume that deg w P 2, and suppose first that wz has either infinitely many zeros or poles or both. Let wz have either a zero or a pole at z = ẑ. Then either there is a cancelation with a zero or pole of some of the coefficients in 3.4, or wz has a pole of order at least at z = ẑ +, or at z = ẑ. Since the coefficients of 3.4 are rational, we can always choose a zero or a pole of wz in such a way that there is no cancelation with the coefficients. Suppose, without loss of generality, that there is a pole of wz at z = ẑ +. By shifting 3.4 up we obtain wz + 2 wz + az + w z + wz + = P z +, wz +, from which it follows that wz has a pole at z = ẑ + 2 of order at least deg w P, and a pole of order at least deg w P 2 at z = ẑ + 3, and so on. The only way that this string of poles with exponential growth in the multiplicity can terminate, or that there can be a drop in the orders of poles, is if there is a cancelation with a suitable zero or pole of a coefficient of 3.4. But since the coefficients are rational and thus have finitely many zeros and poles, and wz has infinitely many zeros or poles, we can choose the starting point ẑ of the iteration from outside a sufficiently large disc in such a way that no cancelation occurs. Thus, nd + ẑ, w deg w P d for all d N, and so λ 2 /w = lim sup r lim sup d lim sup d log log nr, w log r log log nd + ẑ, w logd + ẑ log logdeg w P d logd + ẑ Therefore, ρ 2 w λ 2 /w. Suppose now that wz has finitely many poles and zeros, and that ρ 2 w <. Then =. wz = fz expgz, 3.5 where fz is a rational function and gz is entire. By substituting 3.5 into 3.4, it follows that f fz + e gz+ fz e gz z + az fz + g z = P z, fz expgz. 3.6 Now, since ρ 2 expgz <, it follows from the difference analogue of the lemma on the logarithmic derivatives, [4, Theorem 5.], that T r, e gz+ gz = m r, e gz+ gz = Sr, e g, 6

7 and similarly T r, e gz gz = m r, e gz gz = Sr, e g. Hence, by writing 3.6 in the form e gz fz + e gz+ gz fz e gz gz + az = P z, fz expgz, f z fz + g z and taking Nevanlinna characteristic from both sides, we arrive at the equation deg w P T r, e g = T r, e g + Sr, e g + Olog r. Since deg w P 2 by assumption, this implies that g is a constant. But this means that w is rational, which is a contradiction. Thus ρ 2 w. In our final lemma we consider the case in which Qz, w has a repeated root as a polynomial in w. Lemma 3.3. Let w be a non-rational meromorphic solution of the equation wz + wz + az w z wz = P z, wz wz b z κ ˇQz, wz, 3.7 where a and b are rational functions of z, P z, w and ˇQz, w are polynomials in w and rational in z, and κ is an integer greater than one. Furthermore we assume that w b z, P z, w and ˇQz, w are pairwise co-prime. Then w has hyper-order at least one. Proof. Notice that b z is not a solution of 3.7, even if b 0, thus the first condition of Lemma 2. is satisfied for b. Suppose that ẑ is a zero of wz b z of order p and that neither az, b z nor any of the coefficient functions in P z, w nor ˇQz, w has a zero or a pole at ẑ. We will also require that these coefficient functions do not have zeros or poles at points of the form ẑ + j for a finite number of integers j in this particular case, we take 4 j 4. We will call such a point ẑ a generic zero of order p. We will assume, often without further comment, that in similar situations we are only considering generic zeros. Since the coefficients are rational, when estimating the corresponding unintegrated counting functions, the contribution from the non-generic zeros can be included in a bounded error term, leading to an error term of the type Olog r in the integrated estimates involving T r, w. Now either wz + or wz has a pole of order q κp at z = ẑ, and we suppose without loss of generality that ẑ + is such a pole. Suppose next that deg w P κ + deg w ˇQ. 3.8 Then wz has a pole of order one at ẑ +2 and a pole of order q at ẑ +3. By continuing the iteration, it follows that wz has either a simple pole or a finite value at ẑ + 4. Therefore it may be that wẑ + 4 = b ẑ + 4, and so it is at least in principle possible that wẑ + 5 is a finite value. This can only happen if the order of the zero of wz b z at z = ẑ + 4 is p = q/κ p. But even so, by considering the multiplicities of zeros and poles of w b in the set {ẑ,..., ẑ + 4}, we find that there are 2q + > 2q κp + κp poles of w for p + p zeros of w b. This is the worst case scenario in the sense that if wẑ + 4 b ẑ + 4, or a zero of ˇQz, wz, then ẑ + 5 is a pole of w of order q κp, and we have even more 7

8 poles for every zero of w b. By adding up the contribution from all points ẑ to the corresponding counting functions, it follows that n r, nr + 4, w + O. w b κ Thus both conditions of Lemma 2. are satisfied, and so the hyper-order of w is at least one. Assume now that deg w P κ + deg w ˇQ Suppose again that ẑ is a generic zero of wz b z of order p. Then, as in the case 3.8, wz has a pole of order q κp at either ẑ + or ẑ, say ẑ +. This implies that wz has a pole of order q q at ẑ + 2, and so, the only way that wẑ + 4 can be finite is that wz b z has a zero at ẑ + 3 with multiplicity p = q /κ, or wẑ + 3 is a zero of ˇQz, wz. Even if this would be the case, we have found at least κp + κp poles, taking into account multiplicities, that correspond uniquely to at most p + p zeros of w b. Therefore, we have n r, w b nr + 3, w + O κ by going through all zeros of w b in this way. Lemma 2. thus implies that the hyperorder of w is at least one. Proof of Theorem.. From lemmas 3., 3.2 and 3.3, it follows that deg w P 4 and deg w Q 4 and that Qz, w has only simple roots as a polynomial in w. We will begin with the case in which deg w Q =. We can therefore without loss of generality write the denominator of the right hand side of.3 in the form Qz, w = w b. Assume first that deg w P 3. Let ẑ be a generic zero of wz b z of order p. Then wz has a pole of order at least p at ẑ + or ẑ. We assume without loss of generality that ẑ + is such a pole. Then ẑ + 2 is a pole of order at least 2p and ẑ + 3 is a pole of order at least 4p, and so on. In this case we therefore have n r, w b nr + 2, w + O. 3 Lemma 2. thus implies that the hyper-order of w is at least one. Assume now that Qz, w = w b and deg w P 2. If deg w P = 2, then deg w P = deg w Q + and if deg w P, then deg w R =, thus the assertion.4 holds. Suppose now that the denominator of Rz, wz has at least two simple non-zero rational roots for w as a function of z, say b z 0 and b 2 z 0. Then we may write equation.3 in the form wz + wz + az w z wz = P z, wz wz b zwz b 2 z Qz, wz, 3.0 where P z, wz 0 and Qz, wz 0 are polynomials in wz of at most degree 4 and 2 respectively, with no common factors. Then neither b z, nor b 2 z is a solution of 3.0, and so they satisfy the first condition of Lemma 2.. Let z = ẑ be a generic zero of order p of w b. 8

9 Now, by 3.0, it follows that either wz + or wz has a pole at z = ẑ of order at least p. Without loss of generality we may assume that wz + has such a pole at z = ẑ. Then, by shifting the equation 3.0, we have wz + 2 wz + az + w z + wz + P z +, wz + = wz + b z + wz + b 2 z + Qz +, wz +, 3. which implies that wz + 2 has a pole of order one at z = ẑ provided that deg w P deg w Q We suppose first that 3.2 is valid. By iterating 3.0 one more step, we have wz + 3 wz + + az + 2 w z + 2 wz + 2 P z + 2, wz + 2 = wz + 2 b z + 2wz + 2 b 2 z + 2 Qz + 2, wz Now, if p > then w must be a pole of order at least p at ẑ + 3. Hence, in this case, we can pair up the zero of w b at z = ẑ together with the pole of w at ẑ + without the possibility of a similar sequence of iterates starting from another point, say z = ẑ + 3, and resulting in pairing the pole at ẑ + with another zero of w b, or of w b 2. Therefore, we have found a pole of order at least p which can be uniquely associated with the zero of w b at ẑ. If, on the other hand, p = it may in principle be possible that there is another zero of w b or of w b 2 at z = ẑ + 3 which needs to be paired with the pole of w at z = ẑ + 2. But since now all of the poles in the iteration are simple, we may still pair up the zero of w b at z = ẑ and the pole of w at z = ẑ +. If there is another zero of, say, w b at z = ẑ + 3 such that wẑ + 4 is finite, we can pair it up with the pole of w at z = ẑ + 2. Thus for any p there is a pole of multiplicity at least p which can be paired up with the zero of w b at z = ẑ. We can repeat the argument above for zeros of w b 2 in a completely analogous fashion without any possible overlap in the pairing of poles with the zeros of w b and w b 2. By considering all generic zeros of wz b z, and similarly for wz b 2 z, it follows that n r, + n r, nr +, w + O. 3.4 w b w b 2 Therefore the remaining condition 2.2 of Lemma 2. is satisfied, and so w must be of hyper-order at least one. We consider now the case where the opposite inequality to 3.2 holds, i.e., deg w P > deg w Q + 2. If deg w P = 3, it immediately follows that deg w Q = 2, and so the first part of assertion.4 holds in this case. Now assume that 4 = deg w P > deg w Q + 2 = and suppose that ẑ is a generic zero of wz b z of order p. Then again, by 3.0, either wz + or wz must have a pole at z = ẑ of order at least p, and we suppose as above that wz + has the pole at ẑ. Then, it follows that wz + 2 has a pole of order 9

10 2p, and wz + 3 a pole of order 4p at z = ẑ. Hence we can pair the zero of w b at z = ẑ and the pole of w at z = ẑ + the same way as in the case 3.2. Identical reasoning holds also for the zeros of w b 2, and so 3.4 holds. Lemma 2. therefore yields that w is of hyper-order at least one. Suppose then that 4 = deg w P > deg w Q + 2 = 3, 3.6 and that ẑ is a generic zero of order p of wz b z. Since now deg w Q =, we have Qz, wz = wz b 3 z, where b 3 z b j z for j {, 2} is a rational function of z. Also, it follows by an assumption of the theorem that b 3 0. As before, we see from 3.0 that either wz + or wz has a pole of order at least p at z = ẑ, and we may again suppose that wz + has that pole. If p > then 3. implies that wz + 2 has a pole of order at least p at z = ẑ. Even if w b j has a zero at z = ẑ + 3 for some j {, 2, 3}, we have found at least one pole for each zero of w b j in this iteration sequence, taking multiplicities into account. Hence we can pair the zero of w b at z = ẑ and the pole of w at z = ẑ + the same way as in cases 3.2 and 3.5. However, if p = it may in principle be possible that the pole of the right hand side of 3. at z = ẑ cancels with the pole of the term az + w z + wz + at z = ẑ in such a way that wẑ +2 remains finite. If wẑ +2 b j ẑ +2 for j {, 2, 3}, then it follows from 3.3 that wz +3 has a pole at z = ẑ, and we can pair up the zero of w b at z = ẑ and the pole of w at z = ẑ +. If wẑ +2 = b j ẑ +2 for some j {, 2, 3}, it may happen that also wẑ + 3 stays finite. If all points ẑ such that wẑ = b j ẑ are a part of an iteration sequence of this form, i.e., that wẑ = b j ẑ, wẑ + =, wẑ = b j2 ẑ, j, j 2 {, 2, 3}, then by considering the multiplicities of all zeros of w b j, j {, 2, 3}, we have the inequality n r, + n r, + n r, 2nr +, w + O. w b w b 2 w b 3 As this is the worst case scenario, this estimate remains true in general. Also, we have already noted that neither b, nor b 2 satisfy the equation 3.0. The same is true also for b 3, and so all conditions of Lemma 2. are satisfied. Hence the hyper-order of w is at least one also in the case The proof of Theorem.2 Let z = ẑ be a generic zero of wz, then by.5 there is a pole of wz at z = ẑ + or at z = ẑ or at both points. We need to consider two cases. Suppose first that there is a pole of wz at both points z = ẑ and z = ẑ +. Then, from.5 it follows that there are poles of wz at z = ẑ 2 and z = ẑ + 2. Now, at least in principle we may have wẑ 3 = 0 = wẑ + 3. Hence, in this case we can find at least four poles of wz ignoring multiplicity which correspond to three zeros also ignoring multiplicity of wz and to no other zeros. 0

11 Assume now that there is a pole of wz at only one of the points z = ẑ+ and z = ẑ. Without loss of generality we can then suppose that wz has a pole at z = ẑ + the case where the pole is at z = ẑ is completely analogous. We will begin by showing that we need only consider simple generic zeros of wz. Let N r, /w denote the integrated counting function for the simple zeros of w and let N [p r, /w be the counting function for the zeros of w, which are of order p or higher. Then Nr, /w = N r, /w + N [2 r, /w and N r, = N r, + N w w [2 r, w N 2 N r, w r, w + 2 N [2 r, w + 2 N r, w Hence, using the assumption.6, N r, 2N r, N r, w w w ɛ T r, w N r, w 2 + ɛ T r, w + Sr, w. Thus there are at least /2 + ɛt r, w worth of simple poles of w. So if we consider the case in which the zero of w at ẑ is simple, we have. wz = K + Oz ẑ, K C, α C \ {0} wz = αz ẑ + Oz ẑ 2, wz + = az z ẑ + O, az + wz + 2 = + O, z ẑ az + 2 az wz + 3 = + O z ẑ 4. in a neighborhood of ẑ. If aẑ +2 aẑ 0 then wz +4 = az +3+az +/z ẑ+oz ẑ. Therefore either we have infinitely many points such that az + 2 = az and therefore the rational function a is a constant, or we can find at least four poles for every two simple zeros of w. In the second case it follows that T r, w 2 + ɛn r, + Olog r w 2 N r + 2, w + Olog r + 2ɛ 2 T r, w + Sr, w. + 2ɛ But this implies that T r, w = Sr, w, which is a contradiction. Thus az must be a constant.

12 5 The proof of Theorem.3 Let ẑ be a generic zero of wz of order p. We need to consider two cases. Suppose first that there is a pole of wz at both points z = ẑ and z = ẑ +. Then, even if there are zeros of wz at both z = ẑ 2 and z = ẑ + 2, we can group together three zeros of w ignoring multiplicity with at least four poles of w counting multiplicity. Assume now that there is a pole of wz at only one of the points z = ẑ+ and z = ẑ. Without loss of generality we can then suppose that wz has a pole at z = ẑ + the case where the pole is at z = ẑ is completely analogous. Consider first the case where the zero is simple, and suppose that cz 0. Then, in a neighborhood of ẑ, wz = K + Oz ẑ, K C, wz = αz ẑ + Oz ẑ 2, α C \ {0} wz + = az αz ẑ 2 + bz + cz + K + Oz ẑ, αz ẑ wz + 2 = cz + + Oz ẑ, wz + 3 = az αz ẑ 2 + bz αz ẑ + O, 5. where there can be at most finitely many ẑ such that cẑ + = 0. Hence there are two poles of wz counting multiplicity corresponding to one zero ignoring multiplicity in this case. Assume now that cz 0, wz has a pole at z = ẑ +, and that wẑ is finite. Then, in a neighborhood of ẑ, wz = K + Oz ẑ, K C, wz = αz ẑ + Oz ẑ 2, α C \ {0} wz + = az αz ẑ 2 + bz αz ẑ + O, 2az + wz + 2 = α z ẑ + Oz ẑ 2, az azaz + 2 2az + + az wz + 3 = az 2az + αz ẑ 2 + γz αz ẑ + O, 5.2 where azbz + 2 2az + azbz γz = az 2az + 2az + 2[aza z + az + a z] az 2az If wẑ+3 is a pole of order two, then are at least four poles counting multiplicities in this sequence that can be uniquely grouped with the two zeros of wz ignoring multiplicities. The only way that wz can have a simple pole at z = ẑ + 3 is that aẑ + 2 2aẑ + + aẑ = and γz 0. But in this case from equation.7 it follows that αaz + 3 wz + 4 = + Oz ẑ γz 2

13 for all z in a neighborhood of ẑ, and so wẑ +4 is finite and non-zero with at most finitely many exceptions. Thus we can group together three poles of wz counting multiplicities and two zeros of wz ignoring multiplicities. The only way that wẑ + 3 can be finite is that 5.4 holds together with γ 0. If the order of the zero of wz at z = ẑ is p 2, then there are always at least three poles of wz counting multiplicity for each two zeros of wz ignoring multiplicity in sequence 5. and 5.2. If there are only finitely many zeros ẑ of wz such that 5.4 and γz 0 both hold, then Hence, for any ε > 0, N n r, w 3 nr +, w + O. 4 r, 3 w 4 + ε Nr +, w + Olog r, 2 and so by using [4, Lemma 8.3] to deduce that Nr +, w = Nr, w + Sr, w, we have N r, 3 w 4 + ε T r, w + Sr, w. 2 This is in contradiction with.8, and so there must be infinitely many points ẑ such that 5.4 and γz 0 are both satisfied. The only rational functions az satisfying 5.4 at infinitely many points have the form az = λ + µz, for some constants λ and µ. Equation γz 0 becomes bz + 2 az + 2 bz az = 2az + a z aza z + = µ azaz + 2 az. az + 2 Hence bz = kaz µ, where k is a constant since a and b are assumed to be rational. Acknowledgements The authors thank Bjorn Berntson, Zhibo Huang, and especially Janne Gröhn for their helpful comments on an earlier draft of this paper. We also thank an anonymous referee for suggesting improvements. References [] M. J. Ablowitz, R. Halburd, and B. Herbst, On the extension of the Painlevé property to difference equations, Nonlinearity , [2] B. Grammaticos, A. Ramani, and I. C. Moreira, Delay-differential equations and the Painlevé transcendents, Physica A , [3] R. G. Halburd and R. J. Korhonen, Finite-order meromorphic solutions and the discrete Painlevé equations, Proc. Lond. Math. Soc , [4] R. G. Halburd, R. Korhonen, and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Amer. Math. Soc , no. 8, [5] W. K. Hayman, Meromorphic functions, Clarendon Press, Oxford,

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