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Some results of ϖ-Painlevé difference equation
Advances in Difference Equations volume 2018, Article number: 282 (2018)
Abstract
In this article, we mainly investigate some properties of two types of difference equations
and
1 Introduction
Halburd and Korhonen [4] used Nevanlinna theory to single out difference equations in this form
where \(R(z,Y)\) is rational in O and meromorphic in z, has an admissible meromorphic solution of finite order, then either O satisfies a difference Riccati equation
where \(p(z), q(z) \in S(Y)\), where \(S(Y)\) denotes the field of small functions with respect to Y, or Eq. (1.1) can be transformed to one of the following equations:
where \(\varsigma_{k}, \kappa_{k} \in S(Y)\) are arbitrary finite order periodic functions with period k.
Eqs. (1.3), (1.5), and (1.6) are known alternative forms of difference Painlevé I equation, Eq. (1.8) is a difference Painlevé II, and (1.9) and (1.10) are linear difference equations. Chen and Shon [2, 3] considered some value distribution problems of finite order meromorphic solutions of Eqs. (1.2), (1.5), (1.6), and (1.8). A natural question is: What is the result if we give q-difference analogues of (1.3) and (1.6)? For this question, we consider the following equations:
Theorem 1.1
Let \(Y(z)\) be a transcendental meromorphic solution with zero order of Eq. (1.11) and ξ, o, v be three constants such that ξ, o cannot vanish simultaneously. Then
-
(i)
\(Y(z)\) has infinitely many poles.
-
(ii)
For any finite value B, if \(\xi\neq0\), then \(Y(z)-B\) has infinitely many zeros.
-
(iii)
If \(\xi=0\) and \(Y(z)-A\) has finite zeros, then A is a solution of \(3z^{2}-o-vz=0\).
We assume that the reader is familiar with the basic notions of Nevanlinna theory (see, e.g., [5, 6]).
Theorem 1.2
Let \(c\in\mathbb{C}\setminus\{0\}\), \(|\varpi|\neq1\), and \(V(z)=\frac {X(z)}{B(z)}\) be an irreducible rational function, where \(X(z)\) and \(B(z)\) are polynomials with \(\deg X(z)=x\) and \(\deg B(z)=b\).
-
(i)
Suppose that \(x\geq b\) and \(x-b\) is zero or an even number. If the equation
$$ Y(\varpi z)+Y(z)+Y\biggl(\frac{z}{\varpi}\biggr)=\frac{V(z)}{Y(z)}+c $$(1.13)has an irreducible rational solution \(Y(z)=\frac{I(z)}{J(z)}\), where \(i(z)\) and \(J(z)\) are polynomials with \(\deg i(z)=i\) and \(\deg J(z)=j\), then
$$ i-j=\frac{x-b}{2}. $$ -
(ii)
Suppose that \(x< b\). If Eq. (1.13) has an irreducible rational solution \(Y(z)=\frac{i(z)}{J(z)}\), then \(Y(z)\) satisfies one of the following two cases:
-
(1)
\(Y(z)=\frac{i(z)}{J(z)}=\frac{c}{3}+\frac{T(z)}{D(z)}\), where \(T(z)\) and \(D(z)\) are polynomials with \(\deg T(z)=t\) and \(\deg D(z)=d\), and \(b-x=d-t\).
-
(2)
\(i-j=x-b\).
-
(1)
Theorem 1.3
Let \(Y(z)\) be a transcendental meromorphic solution with zero order of Eq. (1.12) and ξ, o, v be three constants such that ξ, o cannot vanish simultaneously. Then
-
(i)
\(Y(z)\) has infinitely many poles.
-
(ii)
For any finite value B, if \(\xi\neq0\) and \(v\neq0\), then \(Y(z)-B\) has infinitely many zeros.
-
(iii)
If \(\xi=0\) and \(Y(z)-A\) has finite zeros, then A is a solution of \(2z^{2}-oz-v=0\).
Theorem 1.4
Let ξ, o, π be constants with \(\xi\pi\neq0\) and \(|\varpi|\neq 1\). Suppose that a rational function
is a solution of (1.12), where \(F(z)\) and \(U(z)\) are relatively prime polynomials, \(\mu_{0}\neq0\), \(\mu_{1},\ldots, \mu_{m}\), and \(\lambda _{0}\neq0\), \(\lambda_{1},\ldots, \lambda_{n}\) are constants. Then \(n=m+1\) and \(\mu_{0}=-\frac{\pi}{\xi}\lambda_{0}\).
2 Some lemmas
Lemma 2.1
([1])
Let \(Y(z)\) be a non-constant zero order meromorphic solution of
where \(P(z, Y)\) and \(Q(z, Y)\) are ϖ-difference polynomials in \(Y(z)\). If the degree of \(Q(z, Y)\) as a polynomial in \(Y(z)\) and its ϖ-shifts is at most n, then
on a set of logarithmic density 1.
Lemma 2.2
([1])
Let \(Y(z)\) be a non-constant zero order meromorphic solution of
where \(H(z,O)\) is a ϖ-difference polynomial in \(Y(z)\). If \(H(z, Y)\not\equiv0\) for a slowly moving target \(a(z)\), then
on a set of logarithmic density 1.
Lemma 2.3
([7])
Let \(Y(z)\) be a zero order meromorphic function, and \(\varpi\in\mathbb {C}\setminus\{0\}\). Then
3 Proof of Theorem 1.1
(i): Suppose that \(Y(z)\) is a zero order transcendental meromorphic solution of (1.11). By (1.11), we have
where \(P(z, Y)=Y(\varpi z)+Y(z)+Y(\frac{z}{\varpi})\), \(Q(z, Y)=\xi z+o+vY(z)\). Lemma 2.1 implies that
on a set of logarithmic density 1. By the Valiron–Mohon’ko theorem, we obtain that
By Lemma 2.3, we obtain
(3.2), (3.3), and (3.4) imply that
on a set of logarithmic density 1. Hence, \(Y(z)\) has infinitely many poles.
(ii): For any finite value B, and let
Substituting \(Y_{1}(z)=Y(z)-B\) into (3.1), we obtain
Let
If \(\xi\neq0\), by (3.6), we have \(P(z, 0)=3B^{2}-\xi z-o-vB\not\equiv0\). Lemma 2.2 implies that
on a set of logarithmic density 1. Hence
on a set of logarithmic density 1. Hence, \(Y(z)\) has infinitely many finite values.
(iii): If \(\xi=0\) and B is not a solution of \(3z^{2}-o-vz=0\), then \(P(z, 0)=3B^{2}-o-vB\not\equiv0\). Using a similar method as above, we can obtain that
which contradicts the assumption of Theorem 1.1, hence the conclusion holds.
4 Proof of Theorem 1.2
By (1.13) and \(Y(z)=\frac{I(z)}{J(z)}\), we have
Obviously, we have
(i): Suppose first that \(x>b\) and \(x-b\) is an even number. If \(\deg i(z)=i< j=\deg J(z)\), then (4.1)–(4.4) imply that \(x+4j=b+i+3j\), that is, \(0>i-j=x-b>0\). This is impossible.
If \(i=j\), then we use a similar method as above, we can obtain \(0< x-b=i-j=0\), this is impossible. So, \(i>j\). By (4.1), we have \(x+4j=b+2i+2j\), that is, \(i-j=\frac {x-b}{2}\).
(ii): Suppose that \(x< b\). If \(i>j\), then (4.1)–(4.4) yield that \(x+4j=b+2i+2j\), that is, \(0>x-b=2(i-j)>0\), which is a contradiction.
If \(i=j\), then we can assume
where \(\iota_{0} \neq0\), \(T(z)\) and \(D(z)\) are polynomials, and \(\deg T(z)=t<\deg D(z)=d\). Thus, as \(z\rightarrow\infty\), (1.13) and (4.5) imply that
which implies \(\iota_{0}=\frac{c}{3}\). Hence,
Substituting (4.7) into (1.13), we get
Obviously,
Hence, \(b-x=d-t\).
If \(i< j\), by \(i< j\), \(x< b\), (4.1)–(4.4), then we have
5 Proof of Theorem 1.3
(i). Suppose that \(Y(z)\) is a zero order transcendental meromorphic solution of (1.12). By (1.12), we have
Lemma 2.1 implies that
on a set of logarithmic density 1. By the Valiron–Mohon’ko theorem, we get that
By Lemma 2.3, we obtain
(5.2), (5.3), and (5.4) yield that
on a set of logarithmic density 1. Hence, \(Y(z)\) has infinitely many poles.
(ii). For any finite value B, let
Substituting \(Y_{1}(z)=Y(z)-B\) into (5.1), we obtain
Let
By (5.6), we have \(P(z, 0)=2B^{2}-(\xi z+o)B-\pi\).
If \(B=0\) and \(\pi\neq0\), then we obtain that \(P(z, 0)=-v\not\equiv0\).
If \(B\neq0\), then we have \(P(z, 0)=2B^{2}-(\xi z+o)B-v\not\equiv0\) since \(\xi\neq0\). Using a method similar to Theorem 1.1, we can obtain that
on a set of logarithmic density 1. Hence, \(Y(z)\) has infinitely many finite values.
(iii). If \(\xi=0\) and A is not a solution of \(2z^{2}-oz-\pi=0\), then using a method similar to Theorem 1.1, we also obtain that
which contradicts the assumption of Theorem 1.3, hence the conclusion holds.
6 Proof of Theorem 1.4
Assume that (1.12) has a rational solution \(Y(z)\) and has poles \(t_{1}, t_{2},\ldots,t_{k}\). Next, let
be the principal parts of Y at \(t_{j}\), respectively, where \(c_{j\lambda _{j}},\ldots,c_{j1}\) are constants, \(c_{j\lambda_{j}}\neq0\). Hence
where \(\tau_{0},\ldots,\tau_{s}\) are constants. Assume that \(\tau_{s}\neq0\) (\(s\geq1\)). When \(z\rightarrow\infty\),
By (1.12), we have
When \(z\rightarrow\infty\), (6.2), (6.3), and (6.4) imply that
which is a contradiction since \(\tau_{s}\neq0 \) and \(s\geq1\). Assume that \(\tau_{0}\neq0\), as \(z\rightarrow\infty\),
By (6.4) together with (6.5) and (6.6), we obtain that
This is impossible since \(\xi\neq0\) and \(\tau_{0}\neq0\). Hence
where \(\deg F(z)=m<\deg U(z)=n\). Equation (6.7) and (1.12) imply that
Hence we have \(n=m+1\). By (1.12) and \(n=m+1\), we obtain
Since as \(z\rightarrow\infty\), we have
and
we obtain \(\xi\mu_{0}\lambda_{0}+\pi\lambda_{0}^{2}=0\).
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Funding
The work was supported by the NNSF of China (No. 10771121, 11301220, 11401387, 11661052), the NSF of Zhejiang Province, China (No. LQ 14A010007), the NSF of Shandong Province, China (No. ZR2012AQ020), and the Fund of Doctoral Program Research of Shaoxing College of Art and Science (20135018).
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Liu, Y., Zhang, Y. Some results of ϖ-Painlevé difference equation. Adv Differ Equ 2018, 282 (2018). https://doi.org/10.1186/s13662-018-1710-z
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DOI: https://doi.org/10.1186/s13662-018-1710-z