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# Some results on *q*-difference equations

- Junchao Zhang
^{1}, - Gang Wang
^{2}, - Junjie Chen
^{1}Email author and - Rongxiang Zhao
^{3}

**2012**:191

https://doi.org/10.1186/1687-1847-2012-191

© Zhang et al.; licensee Springer 2012

**Received: **16 June 2012

**Accepted: **24 October 2012

**Published: **5 November 2012

## Abstract

In this paper, we consider the *q*-difference analogue of the Clunie theorem. We obtain there is no zero-order entire solution of ${f}^{n}(z)+{({\mathrm{\nabla}}_{q}f(z))}^{n}=1$ when $n\ge 2$; there is no zero-order transcendental entire solution of ${f}^{n}(z)+P(z){({\mathrm{\nabla}}_{q}f(z))}^{m}=Q(z)$ when $n>m\ge 0$; and the equation ${f}^{n}+P(z){\mathrm{\nabla}}_{q}f(z)=h(z)$ has at most one zero-order transcendental entire solution *f* if *f* is not the solution of ${\mathrm{\nabla}}_{q}f(z)=0$, when $n\ge 4$.

**MSC:**30D35, 30D30, 39A13, 39B12.

## Keywords

- uniqueness
*q*-shift*q*-difference equations- entire functions
- zero order
- Nevanlinna theory

## 1 Introduction and main results

*n*at most in the meromorphic function

*f*and its derivatives having meromorphic functions as coefficients. If $T(r)$ is the maximum of the characteristics of the coefficients, then

Later, Clunie’s theorem has been improved into many forms (see [2], pp.39-44) which are valuable tools for studying meromorphic solutions of Painlevé and other non-linear differential equations; see, *e.g.*, [2].

In 2007, Laine and Yang [3] obtained a discrete version of Clunie’s theorem.

**Theorem A**

*Let*

*f*

*be a transcendental meromorphic solution of finite*-

*order*

*ρ*

*of a difference equation of the form*

*where*$U(z,f)$, $P(z,f)$,

*and*$Q(z,f)$

*are difference polynomials such that the total degree*$degU(z,f)=n$

*in*$f(z)$

*and its shifts*,

*and*$degQ(z,f)\le n$.

*Moreover*,

*we assume that*$U(z,f)$

*contains just one term of maximal total degree in*$f(z)$

*and its shifts*.

*Then for each*$\epsilon >0$,

*possibly outside of an exceptional set of finite logarithmic measure*.

*I*be a finite set of multi-indexes $\lambda =({\lambda}_{0},\dots ,{\lambda}_{n})$. A difference polynomial of a meromorphic function $w(z)$ is defined as

*r*tends to infinity outside of an exceptional set

*E*of finite logarithmic measure

*e.g.*, [4]. Moreover, the weight of a difference polynomial (1.1) is defined by

where *λ* and *I* are the same as in (1.1) above. The difference polynomial $P(z,w)$ is said to be homogeneous with respect to $w(z)$ if the degree ${d}_{\lambda}={\lambda}_{0}+\cdots +{\lambda}_{n}$ of each term in the sum (1.1) is non-zero and the same for all $\lambda \in I$.

Recently, Korhonen obtained a new Clunie-type theorem in [4].

**Theorem B**

*Let*$w(z)$

*be a finite*-

*order meromorphic solution of*

*where*$P(z,w)$

*is a homogeneous difference polynomial with meromorphic coefficients*,

*and*$H(z,w)$

*and*$Q(z,w)$

*are polynomials in*$w(z)$

*with meromorphic coefficients having no common factors*.

*If*

*then* $N(r,w)\ne S(r,w)$.

**Theorem C**

*Let*$w(z)$

*be a finite*-

*order meromorphic solution of*

*where*$P(z,w)$

*is a homogeneous difference polynomial with meromorphic coefficients*,

*and*$H(z,w)$

*and*$Q(z,w)$

*are polynomials in*$w(z)$

*with meromorphic coefficients having no common factors*.

*If*

*then for any*$\delta \in (0,1)$,

*where* *r* *goes to infinity outside of an exceptional set of finite logarithmic measure*, *and* $T(r)$ *is the maximum of the Nevanlinna characteristics of the coefficients of* $P(z,w)$, $Q(z,w)$, *and* $H(z,w)$.

*q*-difference equation

where the right-hand side is rational in both arguments, has been widely studied during the last decades; see, *e.g.*, [5–8]. Gundersen *et al.* [9] considered the order of growth of meromorphic solutions of (1.2), from which a *q*-difference analogue of the classical Malmquist theorem [10] is given: if the *q*-difference equation (1.2) admits a meromorphic solution of order zero, then (1.2) reduces to a *q*-difference Riccati equation, *i.e.*, ${deg}_{f}R=1$.

*et al.*[11] treated the functional equation

where $0<|c|<1$ is a complex number, ${a}_{j}(z)$ ($j=0,1,\dots ,n$), and $Q(z)$ are rational functions with ${a}_{0}(z)\not\equiv 0$, ${a}_{1}(z)\equiv 1$. They concluded that all meromorphic solutions of (1.3) satisfy $T(r,f)=O({(logr)}^{2})$. This implies that all meromorphic solutions of (1.3) are of zero order of growth.

*E*is defined by

In particular, we denote by ${S}_{q}(r,f)$ any quantity satisfying ${S}_{q}(r,f)=o(T(r,f))$ for all *r* outside of a set of upper logarithmic density 0 on the set of logarithmic density 1.

In 2009, Liu [12] proved the following the result.

**Theorem D**

*There is no non*-

*constant entire solution with finite order of the non*-

*linear difference equation*

It is well known that ${f}^{n}(z)+{g}^{n}(z)=1$ has no entire solutions when $n\ge 3$ (see [13], Theorem 3), and from Theorem D, we can say there is no non-constant entire solutions with finite order of the equation ${f}^{n}(z)+{({\mathrm{\Delta}}_{c}f(z))}^{n}=1$, when $n\ge 2$.

In this paper, we replace ${\mathrm{\Delta}}_{c}f(z)$ by ${\mathrm{\nabla}}_{q}f(z)$ and get the following result.

**Theorem 1**

*There is no non*-

*constant entire solution with zero order of the non*-

*linear*

*q*-

*difference equation*

*when* $n\ge 2$.

**Theorem 2**

*Let*$P(z)$

*and*$Q(z)$

*be polynomials*,

*and let*

*n*

*and*

*m*

*be integers satisfying*$n>m\ge 0$.

*Then there is no non*-

*constant entire transcendental solution with zero order of the non*-

*linear*

*q*-

*difference equation*

In 2010, Yang and Laine [14] got the following result.

**Theorem E**

*Let*$n\ge 4$

*be an integer*, $M(z,f)$

*be a linear difference polynomial of*

*f*,

*not vanishing identically*,

*and*

*h*

*be a meromorphic function of finite order*.

*Then the difference equation*

*possesses at most one admissible transcendental entire solution of finite order such that all coefficients of* $M(z,f)$ *are small functions of* *f*. *If such a solution* *f* *exists*, *then* *f* *is of the same order as* *h*.

In this paper, we replace difference polynomial by *q*-difference polynomial and get the following result.

**Theorem 3**

*Let*$n\ge 4$

*be an integer*, $\tilde{M}(r,f)$

*be a linear*

*q*-

*difference polynomial of*

*f*,

*not vanishing identically*,

*and*$h(z)$

*be a meromorphic function*.

*Suppose*$f(z)$

*is the solution of*

*q*-

*difference equation*

*If* $f(z)$ *is not the solution of* $\tilde{M}(r,f)=0$, *then equation* (1.7) *possesses at most one transcendental entire solution of zero order*.

## 2 Auxiliary results

The following auxiliary results will be instrumental in proving the theorems.

**Lemma 1** ([5], Theorem 1.2)

*Let*$f(z)$

*be a non*-

*constant zero*-

*order meromorphic function and*$q\in \mathbb{C}\mathrm{\setminus}\{0\}$.

*Then*

**Lemma 2** ([5], Theorem 2.1)

*Let*$f(z)$

*be a non*-

*constant zero*-

*order meromorphic solution of*

*where*$P(z,f)$

*and*$Q(z,f)$

*are*

*q*-

*difference polynomials in*$f(z)$.

*If the degree of*$Q(z,f)$

*as a polynomial in*$f(z)$

*and its*

*q*-

*shifts is at most*

*n*,

*then*

**Lemma 3** ([15], Theorem 1.1)

*Let*$f(z)$

*be a non*-

*constant zero*-

*order meromorphic function and*$q\in \mathbb{C}\mathrm{\setminus}\{0\}$.

*Then*

**Lemma 4** ([15], Theorem 1.3)

*Let*$f(z)$

*be a non*-

*constant zero*-

*order meromorphic function and*$q\in \mathbb{C}\mathrm{\setminus}\{0\}$.

*Then*

**Lemma 5** ([16], Lemma 4)

*If*$T:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$

*is a piecewise continuous increasing function such that*

*then the set*

*has logarithmic density* 0 *for all* ${C}_{1}>1$ *and* ${C}_{2}>1$.

**Lemma 6** *Let* $f(z)$ *be a zero*-*order entire function*, $q\in \mathbb{C}\mathrm{\setminus}\{0\}$, *and* *a* *be a non*-*zero constant*. *If* $f(z)$ *and* ${\mathrm{\nabla}}_{q}f(z)$ *share the set* $\{a,-a\}$ *CM*, *then* $f(z)$ *is a constant*.

*Proof*Since $f(z)$ is an entire function of zero order, and $f(z)$ and ${\mathrm{\nabla}}_{q}f(z)$ share the set $\{a,-a\}$ CM, it is immediate to conclude that

where *k* is a constant.

Since ${h}_{1}$ and ${h}_{2}$ with zero order have no zeros and no poles, both ${h}_{1}$ and ${h}_{2}$ are constants. (2.4) implies that $f(z)$ is a constant.

If ${k}^{2}=1$, from (2.1) we get $f(z)={\mathrm{\nabla}}_{q}f(z)$. According to Lemma 3, it implies that $f(z)$ must be a constant. □

## 3 Clunie theorem for *q*-difference

*q*-difference polynomial case. Let ${d}_{j}\in \mathbb{C}$ for $j=1,\dots ,n$, and let ${I}_{q}$ be a finite set of multi-indexes $\gamma =({\gamma}_{0},\dots ,{\gamma}_{n})$. A difference polynomial of a meromorphic function $w(z)$ is defined as

*r*tends to infinity outside of an exceptional set

*E*of finite logarithmic measure

*q*-shifts of $w(z)$ is denoted by ${deg}_{w}^{q}(P)$, and the order of a zero of $P(z,{x}_{0},{x}_{1},\dots ,{x}_{n})$, as a function of ${x}_{0}$ at ${x}_{0}=0$, is denoted by ${ord}_{0}^{q}(P)$; see,

*e.g.*, [4]. Moreover, the weight of a difference polynomial (1.1) is defined by

where *γ* and ${I}_{q}$ are the same as in (3.1) above. The difference polynomial $P(z,w)$ is said to be homogeneous with respect to $w(z)$, if the degree ${d}_{\gamma}={\gamma}_{0}+\cdots +{\gamma}_{n}$ of each term in the sum (1.1) is non-zero and the same for all $\gamma \in {I}_{q}$.

In this paper, we will obtain the new Clunie theorem for *q*-difference polynomials.

**Theorem 4**

*Let*$w(z)$

*be a zero*-

*order meromorphic solution of*

*where*$P(z,w)$

*is a homogeneous*

*q*-

*difference polynomial with polynomial coefficients*,

*and*$H(z,w)$

*and*$Q(z,w)$

*are polynomials in*$w(z)$

*with polynomial coefficients having no common factors*.

*If*

*then* $N(r,w)\ne {S}_{q}(r,w)$.

*Proof*Since $P(r,w)$ is homogeneous, by Lemma 1 it follows that

which is a contradiction to (3.5). We can conclude that $N(r,w)\ne {S}_{q}(r,w)$. □

**Theorem 5**

*Let*$w(z)$

*be a zero*-

*order meromorphic solution of*

*where*$P(z,w)$

*is a homogeneous*

*q*-

*difference polynomial with polynomial coefficients*,

*and*$H(z,w)$

*and*$Q(z,w)$

*are polynomials in*$w(z)$

*with polynomial coefficients having no common factors*.

*If*

*then*

*Proof*On the one hand, (3.2) and (3.5) imply that

□

## 4 Proof of Theorem 1

From (4.1), we get $f(z)$ and ${\mathrm{\nabla}}_{q}f(z)$ share the set $\{\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\}$ CM. From Lemma 6, we obtain that $f(z)$ is a constant.

## 5 Proof of Theorem 2

*f*is a transcendental entire solution of equation (1.6) with zero order. If ${\mathrm{\nabla}}_{q}f(z)\equiv 0$, then ${f}^{n}(z)=Q(z)$, and the conclusion holds. If ${\mathrm{\nabla}}_{q}f(z)\not\equiv 0$, we have

which is impossible.

## 6 Proof of Theorem 3

*f*and

*g*, which are not the solutions of $\tilde{M}(r,f)=0$ and $\tilde{M}(r,g)=0$, are two distinct zero-order transcendental entire solutions of (1.7), then we can write

*f*,

*g*are zero-order entire functions, we get a contradiction. Therefore, $\frac{f}{g}$ must be a constant. If $f/g=k\ne {\eta}_{j}$,

*k*is a constant, then from (6.1) we have

From (6.9) and (6.2), it is easy to get $\tilde{M}(r,f)=0$ and $\tilde{M}(r,g)=0$, which is impossible.

## Declarations

## Authors’ Affiliations

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