Abstract
In this paper, we mainly investigate properties of finite order transcendental meromorphic solutions of difference Painlevé equations. If f is a finite order transcendental meromorphic solution of difference Painlevé equations, then we get some estimates of the order and the exponent of convergence of poles of \(\Delta f(z)\), where \(\Delta f(z)=f(z+1)-f(z)\).
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1 Introduction and main results
Let f be a function transcendental and meromorphic in the plane. The forward difference is defined in the standard way by \(\Delta f(z)=f(z+1)-f(z)\). In what follows, we assume that the reader is familiar with the basic notions of Nevanlinna value distribution theory (see [1–3]). In addition, we use the notations \(\sigma(f)\) to denote the order of growth of the meromorphic function \(f(z)\), and \(\lambda(f)\) and \(\lambda(\frac{1}{f})\) to denote, respectively, the exponents of convergence of zeros and poles of \(f(z)\).
Painlevé and his colleagues [4] classified all equations of the Painlevé type of the form
where F is rational in y and \(\frac{dy}{dz}\) and (locally) analytic in z. The first two of these are \(P_{\mathrm{I}}\) and \(P_{\mathrm{II}}\):
where α is a constant. The differential Painlevé equations, discovered at the beginning of the last century, have been an important research subject in the field of mathematics and physics.
In the past 18 years, the discrete Painlevé equations became important research problems (see [5]). For example, discrete Painlevé I equations
and discrete Painlevé II equation
where α, β and γ are constants, \(n\in{N}\).
Some results on the existence of meromorphic solutions for certain difference equations were obtained by Shimomura [6] and Yanagihara [7] 30 years ago.
Recently, a number of papers (see [8–20]) focused on complex difference equations and difference analogues of Nevanlinna theory. As the difference analogues of Nevanlinna theory are being investigated, many results on the complex difference equations are rapidly obtained.
Ablowitz et al. [8] looked at a difference equation of the type
where R is rational in both of its arguments, and showed the following theorem.
Theorem A
(see [8])
If the second-order difference equation
where \(a_{i}\) and \(b_{i}\) are polynomials, admits a non-rational meromorphic solution of finite order, then \(\max\{p, q\}\leq2\).
Halburd and Korhonen [15–17] used value distribution theory and a reasoning related to the singularity confinement to single out the difference Painlevé I and II equations from difference equation (1). They obtained that if (1) has a finite order admissible meromorphic solution \(f(z)\), then either f satisfies a difference Riccati equation, or (1) can be transformed by a linear change in f to some classical difference equations, which include difference Painlevé I equations
and difference Painlevé II equation
where a, b and c are constants.
From above, we see that difference Painlevé I and II equations are the development of the differential and discrete Painlevé I and II equations. So they are an important class of difference equations.
Chen and Chen [10] investigated some properties of meromorphic solutions of difference Painlevé I equation and proved the following Theorem B.
Theorem B
(see [10])
Let a, b, c be constants such that \(|a|+|b|\neq0\). If \(f(z)\) is a finite order transcendental meromorphic solution of the difference Painlevé I equation (4), then
-
(i)
\(\lambda(\frac{1}{f})=\lambda(f)=\sigma(f)\);
-
(ii)
if \(p(z)\) is a non-constant polynomial, then \(f(z)-p(z)\) has infinitely many zeros and \(\lambda(f-p)=\sigma(f)\);
-
(iii)
if \(a\neq0\), then \(f(z)\) has no Borel exceptional value;
if \(a=0\), then Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 3z^{2}-cz-b=0\}\).
The main aims of this paper are to consider the properties of finite order transcendental meromorphic solutions of difference Painlevé I and II equations (2)-(5), and we obtain the following results.
Theorem 1.1
Let a, b, c be constants with \(|a|+|b|\neq0\). Suppose that \(f(z)\) is a finite order transcendental meromorphic solution of the difference Painlevé I equation (2). Then
-
(i)
if \(a\neq0\) and \(p(z)\) is a polynomial, then \(f(z)-p(z)\) has infinitely many zeros and \(\lambda(f-p)=\sigma(f)\);
if \(a=0\), then Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 2z^{2}-cz-b=0\}\);
-
(ii)
\(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} ) =\sigma(\Delta f)=\sigma(f)\).
Theorem 1.2
Let a, b, c be constants with \(|a|+|b|+|c|\neq0\). Suppose that \(f(z)\) is a finite order transcendental meromorphic solution of the difference Painlevé II equation (5). Then
-
(i)
if \(a\neq0\) and \(p(z)\) is a nonzero polynomial, then \(f(z)-p(z)\) has infinitely many zeros and \(\lambda(f-p)=\sigma(f)\);
if \(a=0\), then Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 2z^{3}+(b-2)z+c=0\}\);
if \(c\neq0\), then \(\lambda(f)=\sigma(f)\);
-
(ii)
\(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} )=\sigma(\Delta f)=\sigma(f)\).
Remark 1.1
By Theorem 1.2, we conclude that if \(ac\neq0\) and \(f(z)\) is a finite order transcendental meromorphic solution of the difference Painlevé II equation (5), then \(f(z)\) has no Borel exceptional value.
Theorem 1.3
Let a, b, c be constants with \(|a|+|b|+|c|\neq0\). Suppose that \(f(z)\) is a finite order transcendental meromorphic solution of the difference Painlevé I equation (3). Then
-
(i)
\(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} )=\sigma(\Delta f)=\sigma(f)\);
-
(ii)
if \(a=0\), then Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 2z^{3}-bz-c=0\}\).
Theorem 1.4
Let a, b, c be constants with \(|a|+|b|\neq0\). Suppose that \(f(z)\) is a finite order transcendental meromorphic solution of the difference Painlevé I equation (4). Then \(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} )=\sigma(\Delta f)=\sigma(f)\).
Remark 1.2
The following examples show that \(\lambda (\Delta f )=\sigma(f)\) may not hold in above several theorems.
Example 1.1
The meromorphic function \(f(z)=\tan(\frac{\pi}{2}z)\) satisfies the equation
where \(a=c=0\), \(b=-2\) satisfying \(|a|+|b|=2\neq0\). We obtain that
Thus, \(\lambda (\frac{1}{ f} )=\lambda (\frac{1}{\Delta f} )=\sigma(\Delta f)=\sigma(f)=1\) and \(\lambda (\Delta f )=0\neq\sigma(f)\). The solution \(f(z)=\tan(\frac{\pi}{2}z)\) has two Borel exceptional values i and −i satisfying the equation \(2z^{2}-cz-b=2z^{2}+2=0\).
Example 1.2
The meromorphic function \(f(z)=\tan(\frac{\pi}{4}z)\) satisfies the equation
where \(a=c=0\), \(b=4\) and \(|a|+|b|+|c|=4\). We have
Thus, \(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} )=\sigma(\Delta f)=\sigma(f)=1\) and \(\lambda (\Delta f )=0\neq\sigma(f)\). The solution \(f(z)=\tan(\frac{\pi}{4}z)\) has two Borel exceptional values ±i satisfying \(2z^{3}+(b-2)z+c=2z^{3}+2z=0\).
Example 1.3
The meromorphic function \(f(z)=\frac{1}{e^{2\pi iz}+z+1}\) satisfies the equation
where \(a=c=0\), \(b=2\) and \(|a|+|b|+|c|=2\). We have
Thus, \(\lambda (\frac{1}{f} )=\lambda (\frac{1}{\Delta f} )=\sigma(\Delta f)=\sigma(f)=1\) and \(\lambda (\Delta f )=0\). The solution \(f(z)=\frac{1}{e^{2\pi iz}+z+1}\) has a Borel exceptional value 0 satisfying the equation \(2z^{3}+(b-2)z+c=2z^{3}=0\).
2 The proof of Theorem 1.1
We need the following lemmas to prove Theorem 1.1.
Lemma 2.1
Let \(w(z)\) be a non-constant finite order meromorphic solution of
where \(P(z, w)\) is a difference polynomial in \(w(z)\). If \(P(z,a)\not\equiv0\) for a meromorphic function a satisfying \(\lim_{r\rightarrow\infty}\frac{T(r,a)}{T(r,w)}=0\), then
outside of a possible exceptional set of finite logarithmic measure.
Lemma 2.2
Let f be a non-constant finite order meromorphic function. Then
outside of a possible exceptional set of finite logarithmic measure.
Remark 2.1
In [12], Chiang and Feng proved that let f be a meromorphic function with exponent of convergence of poles \(\lambda (\frac{1}{f} )=\lambda<\infty\), \(\eta\neq0\) be fixed, then for each \(\varepsilon>0\),
Lemma 2.3
(see [18])
Let \(f(z)\) be a transcendental meromorphic solution of finite order σ of a difference equation of the form
where \(H(z, f)\) is a difference product of total degree n in \(f(z)\) and its shifts, and where \(P(z,f)\), \(Q(z,f)\) are difference polynomials such that the total degree of \(Q(z,f)\) is ≤n. Then, for each \(\varepsilon>0\),
possibly outside of an exceptional set of finite logarithmic measure.
Lemma 2.4
(Valiron-Mohon’ko, see [2])
Let \(f(z)\) be a meromorphic function. Then, for all irreducible rational functions in f,
with meromorphic coefficients \(a_{i}(z)\) (\(i=0,1,\ldots,m\)), \(b_{j}(z)\) (\(j=0,1,\ldots,n\)), the characteristic function of \(R(z,f(z))\) satisfies
where \(d=\operatorname{deg}_{f}R=\max\{m,n\}\) and \(\Psi(r)=\max_{i,j}\{T(r,a_{i}),T(r,b_{j})\}\).
Lemma 2.5
(see [12])
Let \(f(z)\) be a meromorphic function with order \(\sigma=\sigma(f)<\infty\), and let η be a fixed nonzero complex number, then for each \(\varepsilon>0\), we have
Lemma 2.6
(see [21])
Let \(g:(0,+\infty)\rightarrow R\), \(h:(0,+\infty)\rightarrow R\) be non-decreasing functions. If (i) \(g(r)\leq h(r)\) outside of an exceptional set of finite linear measure, or (ii) \(g(r)\leq h(r)\), \(r \notin{H\cup(0,1]}\), where \(H\subset(1,\infty)\) is a set of finite logarithmic measure, then for any \(\alpha>1\), there exists \(r_{0}>0\) such that \(g(r)\leq h(\alpha r)\) for all \(r>r_{0}\).
Proof of Theorem 1.1
Suppose that \(f(z)\) is a transcendental meromorphic solution of finite order \(\sigma(f)\) of equation (2).
(i) Let \(a\neq0\). Suppose that \(p(z)\) is a polynomial. Set \(g(z)=f(z)-p(z)\). Substituting \(f(z)=g(z)+p(z)\) into (2), we obtain that
It follows from (6) that
By (7), we have
If \(p(z)\equiv0\), then \(P(z,0)=-(az+b)\not\equiv0\). If \(p(z)\equiv\alpha\in{C\backslash\{0\}}\), then
since \(a\neq0\). Suppose that \(p(z)\) is a non-constant polynomial. Set \(p(z)=d_{k}z^{k}+\cdots+d_{0}\), where \(d_{k},\ldots,d_{0}\) are constants, \(d_{k}\neq0\) and \(k\geq1\). Then we obtain that
Thus, by Lemma 2.1, we see that
outside of a possible exceptional set of finite logarithmic measure. Thus,
outside of a possible exceptional set of finite logarithmic measure. Hence, by (8) and Lemma 2.6, we have \(\lambda(f-p)=\sigma(f)\).
If \(a=0\) and \(p(z)=\beta\notin{E}\), then we have
Using a similar method as above, we obtain \(\lambda(f-\beta)=\sigma(f)\). Hence, the Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 2z^{2}-cz-b=0\}\).
(ii) Set \(z=w+1\). Substituting \(z=w+1\) into (2), we obtain that
That is,
Substituting \(f(w+1)=\Delta f(w)+f(w)\) and \(f(w+2)=\Delta f(w+1)+\Delta f(w)+f(w)\) into (9), we have
That is,
Since \(N(R,\Delta f(w+1))\leq N(R+1,\Delta f(w))+o(N(R+1,\Delta f(w)))\), by Lemma 2.2 we see that there is a subset \(E_{1}\subset{(1,\infty)}\) of finite logarithmic measure such that for \(|w|=R\notin{E_{1}\cup[0,1]}\),
By (10) and (11), when \(|w|=R\notin{E_{1}\cup[0,1]}\), we obtain that
That is,
for all \(|w|=R\notin{E_{1}\cup[0,1]}\). By Lemma 2.6 and (12), for any \(\beta_{1}>1\), there exists \(R_{0}>0\) such that
for all \(R>R_{0}\). By (13), we have
By Remark 2.1 and (14), we have
and
Hence, we get
By (2), we obtain that
By (16) and Lemma 2.3, we see that for any given \(\varepsilon>0\), there is a subset \(E_{2}\subset{(1,\infty)}\) of finite logarithmic measure such that for \(|z|=r\notin{E_{2}\cup[0,1]}\),
By Lemma 2.2, we see that there is a subset \(E_{3}\subset{(1,\infty)}\) of finite logarithmic measure such that for \(|z|=r\notin{E_{3}\cup[0,1]}\),
By Lemma 2.4 and (2), we see that
since \(|a|+|b|\neq0\). By (17)-(19), when \(|z|=r\notin{E_{3}\cup E_{2}\cup[0,1]}\), we have
By Lemma 2.6 and (20), for any \(\beta_{2}>1\), there exists \(r_{0}>0\) such that
for all \(r>r_{0}\). Thus, we get
By (15) and (21), we have \(\lambda (\frac{1}{\Delta f} )\geq\lambda (\frac{1}{f} )\geq\sigma(f)\). And we have \(\sigma (\Delta f )\leq\sigma(f)\) from Lemma 2.5. Hence,
□
3 The proof of Theorem 1.2
Suppose that \(f(z)\) is a transcendental meromorphic solution of finite order \(\sigma(f)\) of equation (5).
(i) Suppose that \(a\neq0\) and \(p(z)\) is a nonzero polynomial. Set \(g_{1}(z)=f(z)-p(z)\). Substituting \(f(z)=g_{1}(z)+p(z)\) into (5), we obtain that
It follows from (22) that
By (23), we have
If \(p(z)\equiv\alpha\in{C\backslash\{0\}}\), then
since \(a\neq0\). Suppose that \(p(z)\) is a non-constant polynomial. Set \(p(z)=d_{k}z^{k}+\cdots+d_{0}\), where \(d_{k},\ldots,d_{0}\) are constants, \(d_{k}\neq0\) and \(k\geq1\). Then we obtain that
Thus, by Lemma 2.1, we see that
outside of a possible exceptional set of finite logarithmic measure. Thus,
outside of a possible exceptional set of finite logarithmic measure. Hence, by (24) and Lemma 2.6, we have \(\lambda(f-p)=\sigma(f)\).
If \(a=0\) and \(p(z)=\beta\notin{E}\), then we have
Using a similar method as above, we obtain \(\lambda(f-\beta)=\sigma(f)\). Hence, the Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 2z^{3}+(b-2)z+c=0\}\).
If \(c\neq0\), then by (5) we have
Hence, we have
Using a similar method as above, we obtain \(\lambda(f)=\sigma(f)\).
(ii) Set \(z=w+1\). Substituting \(z=w+1\) into (5), we obtain that
That is,
Substituting \(f(w+1)=\Delta f(w)+f(w)\) and \(f(w+2)=\Delta f(w+1)+\Delta f(w)+f(w)\) into (25), we have
Thus, we obtain that
where
Since \(N(R,\Delta f(w+1))\leq N(R+1,\Delta f(w))+o(N(R+1,\Delta f(w)))\), by Lemma 2.2 we see that there is a subset \(H_{1}\subset{(1,\infty)}\) having finite logarithmic measure such that for \(|w|=R\notin{H_{1}\cup[0,1]}\),
By (26) and (27), when \(|w|=R\notin{H_{1}\cup[0,1]}\), we obtain that
That is,
for \(|w|=R\notin{H_{1}\cup[0,1]}\). By Lemma 2.6 and (28), for any \(\beta_{1}>1\), there exists \(R_{0}>0\) such that
for all \(R>R_{0}\). Thus, we have
By the same reasoning as in Theorem 1.1(ii), we get
By (5), we obtain that
By (31) and Lemma 2.3, we see that for any given \(\varepsilon>0\), there is a subset \(H_{2}\subset{(1,\infty)}\) having finite logarithmic measure such that for \(|z|=r\notin{H_{2}\cup[0,1]}\),
By Lemma 2.2, we see that there is a subset \(H_{3}\subset{(1,\infty)}\) of finite logarithmic measure such that for \(|z|=r\notin{H_{3}\cup[0,1]}\),
By Lemma 2.4 and (5), we see that
since \(|a|+|b|+|c|\neq0\). By (32)-(34), when \(|z|=r\notin{H_{3}\cup H_{2}\cup[0,1]}\), we have
By Lemma 2.6 and (35), for any \(\beta_{2}>1\), there exists \(r_{1}>0\) such that
for all \(r>r_{1}\). Thus, we get
By (30) and (36), we see that \(\lambda (\frac{1}{\Delta f} )\geq\lambda (\frac{1}{f} )\geq\sigma(f)\). And we have \(\sigma (\Delta f )\leq\sigma(f)\) from Lemma 2.5. Hence,
4 Proofs of Theorems 1.3 and 1.4
Proof of Theorem 1.3
Suppose that \(f(z)\) is a transcendental meromorphic solution of finite order \(\sigma(f)\) of equation (3).
(i) Using the same methods as in the proofs of Theorems 1.1(ii) and 1.2(ii), we have
where
Set \(|w|=R\). Since \(N(R,\Delta f(w+1))\leq N(R+1,\Delta f(w))+o(N(R+1,\Delta f(w)))\), by Lemma 2.2 we see that
outside of a possible exceptional set of finite logarithmic measure. By (37) and (38), we obtain that
outside of a possible exceptional set of finite logarithmic measure. By (39) and Lemma 2.6, we have
By Remark 2.1 and (40), we obtain that
and
Hence, we get
By (3), we obtain that
By (42) and Lemma 2.3, we see that
outside of a possible exceptional set of finite logarithmic measure. By Lemma 2.2, we get
outside of a possible exceptional set of finite logarithmic measure.
If \(c\neq0\), by Lemma 2.4 and (3), we see that
If \(c=0\), by the same reason, we have
since \(|a|+|b|+|c|=|a|+|b|\neq0\). By (43)-(46), we have
outside of a possible exceptional set of finite logarithmic measure. By Lemma 2.6 and (47), we get
By (41) and (48), we have \(\lambda (\frac{1}{\Delta f} )\geq\lambda (\frac{1}{f} )\geq\sigma(f)\). And we have \(\sigma (\Delta f )\leq\sigma(f)\) from Lemma 2.5. Hence,
(ii) Let \(a=0\) and \(p(z)=\beta\notin{E}\). Using the same methods as in the proof of Theorem 1.1(i), we have
Thus, we obtain \(\lambda(f-\beta)=\sigma(f)\). Hence, the Borel exceptional values of \(f(z)\) can only come from the set \(E=\{z \mid 2z^{3}-bz-c=0\}\). □
Proof of Theorem 1.4
Suppose that \(f(z)\) is a transcendental meromorphic solution of finite order \(\sigma(f)\) of equation (4). Set \(z=w+1\). Using the same method as in the proof of Theorem 1.3(i), we have
By the same reason as that in Theorem 1.3(i), when \(|w|=R\), we have
possibly outside of an exceptional set of finite logarithmic measure.
By Lemma 2.6 and (49), we obtain that \(\lambda (\frac{1}{\Delta f(w)} )\geq\lambda (\frac{1}{f(w)} )\). By the same method as above, we have \(\lambda (\frac{1}{\Delta f(z)} )\geq\lambda (\frac{1}{f(z)} )\). And we see that \(\lambda (\frac{1}{f(z)} )=\sigma(f(z))\) from Theorem B and \(\sigma (\Delta f )\leq\sigma(f)\) from Lemma 2.5. Hence, we have
□
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Acknowledgements
The authors thank the referee for his/her valuable suggestions. This work is supported by the State Natural Science Foundation of China (No. 61462016), the Science and Technology Foundation of Guizhou Province (Nos. [2014]2125; [2014]2142) and the Doctoral Foundation of Guizhou Normal University (No. 11904-0514021).
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C-WP completed the main part of this article, C-WP and Z-XC corrected the main theorems. All authors read and approved the final manuscript.
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Peng, CW., Chen, ZX. On properties of meromorphic solutions for difference Painlevé equations. Adv Differ Equ 2015, 123 (2015). https://doi.org/10.1186/s13662-015-0463-1
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DOI: https://doi.org/10.1186/s13662-015-0463-1