Some results on q-difference equations

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.

1 Introduction and main results

It is well known that Clunie’s theorem (see [1], Lemma 1; also see [2], p.39, Lemma 2.4.1) is a useful tool in studying complex differential equations. It states that $Q_n(f)$ is a polynomial of total degree 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

$$\frac{1}{2\pi }{\int }_{|f|>1}{log}^{+}|f^{-n}{Q}_n(f)|d\phi =O(logr+logT(r,f)+T(r))\text{n.e. as }r\to \mathrm{\infty }.$$

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

$$U(z,f)P(z,f)=Q(z,f),$$

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 $\epsilon >0$,

$$m(r,P(z,f))=O\left(r^{\rho -1+\epsilon }\right)+S(r,f)$$

possibly outside of an exceptional set of finite logarithmic measure.

Now let us introduce some notation. Let $c_j\in \mathbb{C}$ for $j=1,\dots ,n$, and let 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

$$\begin{array}{rcl}P(z,w)& =& P(z,w(z),w(z+c_1),\dots ,w(z+c_n))\\ =& \sum _{\lambda \in I}{a}_{\lambda }(z)w{(z)}^{{\lambda }_0}w{(z+c_1)}^{{\lambda }_1}\cdots w{(z+c_n)}^{{\lambda }_n},\end{array}$$

(1.1)

where the coefficients $a_{\lambda }(z)$ are small with respect to $w(z)$ in the sense that $T(r,a_{\lambda })=o(T(r,w))$ as r tends to infinity outside of an exceptional set E of finite logarithmic measure

$$\underset{r\to \mathrm{\infty }}{lim}{\int }_{E\cap [1,r)}\frac{dt}{t}<\mathrm{\infty }.$$

The total degree of $P(z,w)$ in $w(z)$ and in the shifts of $w(z)$ is denoted by ${deg}_w(P)$, 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 ${ord}_0(P)$; see, e.g., [4]. Moreover, the weight of a difference polynomial (1.1) is defined by

$$K(P)=\underset{\lambda \in I}{max}\left\{\sum _{j=1}^n{\lambda }_j\right\},$$

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

$$H(z,w)P(z,w)=Q(z,w),$$

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

$$max\{{deg}_w(H),{deg}_w(Q)-{deg}_w(P)\}>min\{{deg}_w(P),{ord}_0(Q)\}-{ord}_0(P),$$

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

Theorem C Let $w(z)$ be a finite-order meromorphic solution of

$$H(z,w)P(z,w)=Q(z,w),$$

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

$$2K(P)\le max\{{deg}_w(Q),{deg}_w(H)+{deg}_w(P)\}-min\{{deg}_w(P),{ord}_0(Q)\},$$

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

$$m(r,w)=o\left(\frac{T(r,w)}{r^{\delta }}\right)+O(T(r)),$$

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 $H(z,w)$.

The non-autonomous Schröder q-difference equation

$$f(qz)=R(z,f(z)),$$

(1.2)

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., ${deg}_{f}R=1$.

Bergweiler et al. [11] treated the functional equation

$$\sum _{j=0}^n{a}_j(z)f\left(c^{j}z\right)=Q(z),$$

(1.3)

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.

Let us recall ${\mathrm{\Delta }}_{c}f(z)$ of a meromorphic function in the whole plane ℂ is given by

$${\mathrm{\Delta }}_{c}f(z)=f(z+c)-f(z),$$

while

$${\mathrm{\nabla }}_{q}f(z)=f(qz)-f(z)$$

denotes ${\mathrm{\nabla }}_{q}f(z)$ of a meromorphic function in the whole plane ℂ, where $c\in \mathbb{C}\mathrm{\setminus }\{0\}$ and $q\in \mathbb{C}\mathrm{\setminus }\{0,1\}$. The upper logarithmic density of E is defined by

$$\overline{logdens(E)}=\underset{r\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\frac{{\int }_{E\cup [1,r]}\frac{dt}{t}}{logr}.$$

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

$$f^2(z)+{({\mathrm{\Delta }}_{c}f(z))}^2=1.$$

(1.4)

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

$$f^n(z)+{({\mathrm{\nabla }}_{q}f(z))}^n=1,$$

(1.5)

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

$$f^n(z)+P(z){({\mathrm{\nabla }}_{q}f(z))}^m=Q(z).$$

(1.6)

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

$$f^n+M(z,f)=h$$

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

$$f^n+N(r,f)=h(z).$$

(1.7)

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

$$m(r,\frac{f(qz)}{f(z)})=S_q(r,f).$$

Lemma 2 ([5], Theorem 2.1)

Let $f(z)$ be a non-constant zero-order meromorphic solution of

$$f^n(z)P(z,f)=Q(z,f),$$

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

$$m(r,P(z,f))=S_q(r,f).$$

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

$$T(r,f(qz))=T(r,f)+S_q(r,f).$$

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

$$N(r,f(qz))=N(r,f)+S_q(r,f).$$

Lemma 5 ([16], Lemma 4)

If $T:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is a piecewise continuous increasing function such that

$$\underset{r\to \mathrm{\infty }}{lim}\frac{logT(r)}{logr}=0,$$

then the set

$$E:=\{r:T(C_{1}r)\ge C_{2}T(r)\}$$

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

$$\frac{(f(z)-a)(f(z)+a)}{({\mathrm{\nabla }}_{q}f(z)+a)({\mathrm{\nabla }}_{q}f(z)-a)}\equiv k^2,$$

(2.1)

where k is a constant.

If $k^2\ne 1$, let

$$h_1(z):=f(z)+k{\mathrm{\nabla }}_{q}f(z),$$

(2.2)

and

$$h_2(z):=f(z)-k{\mathrm{\nabla }}_{q}f(z).$$

(2.3)

Then $h_1(z)$ and $h_2(z)$ are entire functions, and

$$f(z)=\frac{h_1(z)+h_2(z)}{2},{\mathrm{\nabla }}_{q}f(z)=\frac{h_1(z)-h_2(z)}{2k}.$$

(2.4)

From (2.1)-(2.3), we have

$$h_1(z)h_2(z)=f^2(z)-k^2{({\mathrm{\nabla }}_{q}f(z))}^2=(1-k^2)a^2.$$

(2.5)

From (2.2), (2.3), and (2.5), we obtain

$$N(r,\frac{1}{h_1})=N(r,\frac{1}{h_2})=0.$$

(2.6)

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

Let us consider the 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

$$\begin{array}{rcl}P(z,w)& =& P(z,w(qz),w\left(q^{2}z\right),\dots ,w\left(q^{n}z\right))\\ =& \sum _{\gamma \in I_q}{a}_{\gamma }(z)w{(z)}^{{\gamma }_0}w{(qz)}^{{\gamma }_1}\dots w{\left(q^{n}z\right)}^{{\gamma }_n},\end{array}$$

(3.1)

where the coefficients $a_{\gamma }(z)$ are small with respect to $w(z)$ in the sense that $T(r,a_{\gamma })=o(T(r,w))$ as r tends to infinity outside of an exceptional set E of finite logarithmic measure

$$\underset{r\to \mathrm{\infty }}{lim}{\int }_{E\cap [1,r)}\frac{dt}{t}<\mathrm{\infty }.$$

The total degree of $P(z,w)$ in $w(z)$ and in the 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

$$K_q(P)=\underset{\gamma \in I_q}{max}\left\{\sum _{j=1}^n{\gamma }_j\right\},$$

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

$$H(z,w)P(z,w)=Q(z,w),$$

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

$$max\{{deg}_w^q(H),{deg}_w^q(Q)-{deg}_w^q(P)\}>min\{{deg}_w^q(P),{ord}_0^q(Q)\}-{ord}_0^q(P),$$

then $N(r,w)\ne S_q(r,w)$.

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

$$m(r,\frac{P(r,w)}{w^{{deg}_w^q(P)}})=S_q(r,w).$$

(3.2)

Moreover, Mohon’ko’s theorem (see [17], Theorem 1.13) implies that

$$T(r,\frac{P(r,w)}{w^{{deg}_w^q(P)}})=d_{w}T(r,w)+O(logr),$$

(3.3)

where

$$d_m=max\{{deg}_w^q(Q),{deg}_w^q(H)+{deg}_w^q(P)\}-min\{{deg}_w^q(P),{ord}_0^q(Q)\}.$$

(3.4)

According to (3.2), (3.3), (3.4), and the assumption of Theorem 4, it follows that

$$N(r,\frac{P(r,w)}{w^{{deg}_w^q(P)}})\ge (1+{deg}_w^q(P)-{ord}_0^q(P))T(r,w)+S_q(r,w).$$

(3.5)

This contradicts the assertion of Theorem 4 that $N(r,w)=S_q(r,w)$. Let us denote $q_{max}=max\{|q_j|,(j=1,\dots ,n)\}$, by Lemma 5 we will obtain that

$$\begin{array}{rcl}N(r,\frac{P(r,w)}{w^{{ord}_0^q(P)}})& \le & ({deg}_w^q(P)-{ord}_0^q(P))N(q_{max}r,w)\\ \le & ({deg}_w^q(P)-{ord}_0^q(P)+o(1))N(r,w)+S_q(r,w)\\ =& S_q(r,w).\end{array}$$

Therefore,

$$\begin{array}{rcl}N(r,\frac{P(r,w)}{w^{{deg}_w^q(P)}})& \le & N(r,\frac{P(r,w)}{w^{{ord}_0^q(P)}})+N(r,\frac{1}{w^{{deg}_w^q(P)-{ord}_0^q(P)}})\\ =& N(r,\frac{1}{w^{{deg}_w^q(P)-{ord}_0^q(P)}})+S_q(r,w)\\ \le & T(r,\frac{1}{w^{{deg}_w^q(P)-{ord}_0^q(P)}})+S_q(r,w)\\ =& ({deg}_w^q(P)-{ord}_0^q(P))T(r,w)+S_q(r,w),\end{array}$$

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

$$H(z,w)P(z,w)=Q(z,w),$$

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

$$2K_q(P)\le max\{{deg}_w^q(Q),{deg}_w^q(H)+{deg}_w^q(P)\}-min\{{deg}_w^q(P),{ord}_0^q(Q)\},$$

then

$$m(r,w)=S_q(r,w).$$

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

$$N(r,\frac{P(r,w)}{w^{{deg}_w^q(P)}})=d_{w}T(r,w)+S_q(r,w).$$

(3.6)

On the other hand, by Lemma 4, we can obtain that

$$N(r,\frac{P(r,w)}{w^{{deg}_w^q(P)}})\le K_q(P)(2T(r,w)-m(r,w))+S_q(r,w).$$

(3.7)

(3.6) and (3.7) show that

$$m(r,w)=S_q(r,w).$$

 □

4 Proof of Theorem 1

If $n\ge 3$, there are no non-constant entire solutions of equation (1.5) according to [13]. Let us consider $n=2$, that is,

$$f^2(z)+{({\mathrm{\Delta }}_{q}f(z))}^2=1.$$

(4.1)

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

Suppose that 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

$$f^{n-1}f=Q(z)-P(z)\frac{({\mathrm{\Delta }}_{q}f(z))}{f^m}{f}^m.$$

From Lemma 1, Lemma 2, and the condition $n>m$, we have

$$T(r,f)=m(r,f)=S_q(r,f),$$

which is impossible.

6 Proof of Theorem 3

Assume now, contrary to the assertion, that 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

$$f^n+\tilde{M}(r,f)=g^n+\tilde{M}(r,g).$$

(6.1)

From (6.1), we obtain

$$f^n-g^n=\tilde{M}(r,g)-\tilde{M}(r,f).$$

(6.2)

Therefore, we have

$$F:=\frac{f^n(z)-g^n(z)}{f(z)-g(z)}=\prod _{j=1}^{n-1}(f-{\eta }_{j}g)=-\frac{\tilde{M}(r,f)-\tilde{M}(r,g)}{f(z)-g(z)}$$

(6.3)

is an entire function, and ${\eta }_1,\dots ,{\eta }_{n-1}$ are distinct roots ≠1 of the equation $z^n=1$. Hence, $N(r,\frac{\tilde{M}(r,f)-\tilde{M}(r,g)}{f(z)-g(z)})=0$. From Lemma 1, we get

$$T(r,\frac{\tilde{M}(r,f)-\tilde{M}(r,g)}{f(z)-g(z)})=S_q(r,f-g).$$

(6.4)

If $\frac{f}{g}$ is not a constant, (6.3) implies that

$$(n-1)T(r,\frac{f}{g})=T(r,g^{n-1})+S_q(r,f-g).$$

(6.5)

Thus,

$$\begin{array}{rcl}\sum _{j=1}^{n-1}N(r,\frac{1}{\frac{f}{g}-{\eta }_j})& =& N(r,\frac{f-g}{\tilde{M}(r,f)-\tilde{M}(r,g)})\\ \le & T(r,\frac{f-g}{\tilde{M}(r,f)-\tilde{M}(r,g)})\\ =& S_q(r,f-g).\end{array}$$

From the second fundamental theorem, we have

$$(n-3)T(r,\frac{f}{g})\le \sum _{j=1}^{n-1}N(r,\frac{1}{\frac{f}{g}-{\eta }_j})+O(logr)=S_q(r,f-g).$$

(6.6)

From (6.6), (6.5), and 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

$$(k^n-1)g^n=\tilde{M}(r,(1-k)g).$$

(6.7)

From Lemma 2 and (6.7), we get a contradiction. Thus, $f/g={\eta }_j$ for some $j=1,\dots ,n-1$ and

$$f(z)={\eta }_{j}g(z).$$

(6.8)

This implies that

$$f^n=g^n$$

(6.9)

and

$$\tilde{M}(r,f)={\eta }_j\tilde{M}(r,g).$$

(6.10)

From (6.9) and (6.2), we obtain

$$\tilde{M}(r,f)=\tilde{M}(r,g).$$

(6.11)

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.