All meromorphic solutions of Fermat-type functional equations

In 1637, Fermat [12] stated the conjecture (which is known as Fermat’s last theorem) that the equation $x^{m}+y^{m}=1$ cannot have positive rational solutions if $m>2$. Since then, the equation has been a subject of intense and often heated discussions. In 1995, Wiles [40, 42] proved the profound conjecture.

In 1927, Montel in [38] initially considered the functional equations

which can be regarded as the analogous of Fermat diophantine equations $x^{m}+y^{m}=1$ over function fields. He showed that all the entire solutions $f(z)$ and $g(z)$ of (1.1) must be constant if $m\geq 3$, see also Jategaonka [25]. The follow-up works were given by Baker in [1] and Gross in [15], respectively, they generalized Montel’s result by proving that (1.1) does not have nonconstant meromorphic solutions when $m>4$ and described nonconstant meromorphic solutions for $n=2,~{}3$. In 1970, Yang [45] considered the more general functional equations

and derived that $f^{m}(z)+g^{n}(z)=1$ does not admit nonconstant entire solutions if $\frac{1}{n}+\frac{1}{m}<1$. Since then, (1.2) has been studied in various settings, see [14, 30, 31, 45]. For the convenience, some results can be stated as follows. (see e.g., [8, Proposition 1], [1, 16, 44]).

(i) If $n=m=3$, then $f(z)=\frac{\frac{1}{2}\left\{1+\frac{\wp^{\prime}(h(z))}{\sqrt{3}}\right\}}{\wp(h(z))}$ and $g(z)=\frac{\frac{\eta}{2}\left\{1-\frac{\wp^{\prime}(h(z))}{\sqrt{3}}\right\}}{\wp(h(z))}$ for nonconstant entire function $h$, where $\eta^{3}=1$ and $\wp$ denotes the Weierstrass $\wp$-function satisfying $(\wp^{\prime})^{2}\equiv 4\wp^{3}-1$ after appropriately choosing its periods.

(ii) If $n=2$ and $m=3$, then $f(z)=i\wp^{\prime}(h(z))$ and $g(z)=\eta\sqrt[3]{4}\wp(h(z)),$ where $h(z)$ is any nonconstant entire function.

(iii)If $m=2$ and $n=4$ , then $f(z)=sn^{\prime}(h(z))$ and $g(z)=sn(h(z))$, where $h(z)$ is any nonconstant entire function and $sn$ is Jacobi elliptic function satisfying ${sn^{\prime}}^{2}=1-sn^{4}$.

In 1989, Yanagihara [44] considered the existence of meromorphic solutions of Fermat-type functional equation in another direction. In fact, Yanagihara obtained that $f^{3}(z)+f^{3}(z+c)=1$ does not admit nonconstant meromorphic functions of finite order. This result was also derived by Lü-Han in [22] by making use of the difference analogue of the logarithmic derivative lemma of finite order meromorphic functions, which was established by Halburd and Korhonen [19], Chiang and Feng [9], independently. In 2014, this difference analogue was improved by Halburd, Korhonen and Tohge [21] to meromorphic functions of hyper-order strictly less than 1. Here, the order and hyper-order of a meromorphic function $f$ are defined as

where $T(r,f)$ denotes the Nevanlinna characteristic function of $f$.

With the help of these results, Korhonen and Zhang in [27] generalize Yanagihara’s theorem to meromorphic solutions of hyper-order strictly less than 1 as follows.

Recently, Lü and his co-workers [7, 22, 36] also considered this kind of problems, and their results can be stated as follows.

In recent years, Fermat-type difference as well as differential-difference equations have been studied extensively. As a result, successively a lot of investigations have done by many scholars in this direction, see e.g., [33, 34]. We note that most of above results, including Theorems B and C, were obtained under the condition that the solutions is of hyper-order strictly less than 1, since the difference analogue of the logarithmic derivative lemma is needed in proofs of them. So, it is natural to ask what will happen if the hyper-order condition is omitted. We find that the conclusions of some previous theorems maybe invalid. In [22], Han offered an example to show this point as follows.

Example 1. Let $c=\pi i$ and $\alpha,\beta$ be fixed constants satisfying $e^{\alpha c}=1$. Consider $f(z)=\frac{1}{2}\frac{1+\frac{\wp^{\prime}(h(z))}{\sqrt{3}}}{\wp(h(z))}e^{\frac{\alpha z+\beta}{3}}$, where $h(z)=e^{z}$. Then, a routine computation leads to $f(z+c)=\frac{\eta}{2}\frac{1-\frac{\wp^{\prime}(h(z))}{\sqrt{3}}}{\wp(h(z))}e^{\frac{\alpha z+\beta}{3}}$, where $\eta=e^{\frac{\alpha c}{3}}$. Further, one has

which implies that the above equation admits meromorphic solution $f(z)$. Obviously, $f(z)$ is not the form $de^{\frac{\alpha z+\beta}{3}}$ and the hyper-order of $f$ is $\rho_{2}(f)=1$.

Therefore, if the hyper-order condition is omitted, the Fermat-type difference equation may admit the other type of meromorphic solutions. In this paper, we pay attention to above question. Due to properties of elliptic functions, we describe the forms of all meromorphic solutions of some Fermat-type difference equations.

Before giving our main results, we introduce the Weierstrass $\wp$-function.

The Weierstrass $\wp$-function is elliptic (also doubly periodic) function with periods $\omega_{1}$ and $\omega_{2}$ (${\sf Im}\frac{\omega_{1}}{\omega_{2}}\neq 0$) which is defined as

and satisfies, after appropriately choosing $\omega_{1}$ and $\omega_{2}$,

The period of $\wp$ span the lattice $\mathbb{L}=\omega_{1}\mathbb{Z}\oplus\omega_{2}\mathbb{Z}$. We also denote

Obviously, all the points in $\mathbb{L}$ are the poles and periods of $\wp$. Suppose that $D$ is the parallelogram with vertices at 0, $\omega_{1}$, $\omega_{2}$, $\omega_{1}+\omega_{2}$. Note that the order of $\wp$ is 2. Here, the order is the number of poles of $\wp$ or the number of zeros of $\wp-a$ ($a\in\mathbb{C}$) in the parallelogram. Together with $(\wp^{\prime})^{2}=4\wp^{3}-1$, one gets that $\wp$ has two distinct zeros in $D$, say $\theta_{1}$ and $\theta_{2}$. In view of $\wp(z)=0\Leftrightarrow\wp^{\prime}(z)=\pm i$, without loss of generality, throughout the paper, we assume that $\wp^{\prime}(\theta_{1})=-i$ and $\wp^{\prime}(\theta_{2})=i$.

Here, for two meromorphic functions $f,~{}g$ and two points $a,~{}b\in\mathbb{C}\cup\{\infty\}$, the notation $f(z)-a=0\Rightarrow g(z)-b=0$ means that all the zeros of $f-a$ are the zeros of $g(z)-b$. And the notation $f(z)-a=0\Leftrightarrow g(z)-b=0$ means that $f-a$ and $g-b$ have the same zeros.

In the present paper, more generally, we characterize meromorphic solutions of Fermat-type functional equations as follows.

(1). $h(L(z))=Ah(z)+\theta_{1}+\tau_{1}$ with $\tau_{1}\in\mathbb{L}$;

(2). $h(L(z))=Ah(z)+\theta_{2}+\tau_{2}$ with $\tau_{2}\in\mathbb{L}$;

(3). $h(L(z))=Ah(z)+\tau_{3}$ with $\tau_{3}\in\mathbb{L}$, where $A$ is a constant with $A^{3}=-1$ and $A\tau\in\mathbb{L}$ for any period of $\wp(z)$.

It is pointed out that if $m\neq n$ in Proposition 2, the conclusion $|q|=1$ maybe invalid, as shown by the following example.

Suppose $m\neq n$ and $q^{2}=\frac{m}{n}$. Consider $g(z)=z^{2}+a$ with a constant $a$ satisfying $(1-\frac{m}{n})a=nLn\frac{B}{A}$. Then, a calculation yields $g(qz)=\frac{m}{n}g(z)+nLn\frac{B}{A}$ and $f(z)=Ae^{\frac{g(z)}{n}}$ is a solution of (1.5). But $|q|\neq 1$.

Let’s turn back to the Fermat-type difference equations. Suppose $L(z)=z+c$ with $c\neq 0$. Then, by Proposition 1, we obtain the following result.

(1). $h(z+c)=Ah(z)+\theta_{1}+\tau_{1}$ with $\tau_{1}\in\mathbb{L}$;

(2). $h(z+c)=Ah(z)+\theta_{2}+\tau_{2}$ with $\tau_{2}\in\mathbb{L}$;

(3). $h(z+c)=Ah(z)+\tau_{3}$ with $\tau_{3}\in\mathbb{L}$, where $A$ is a constant with $A^{3}=-1$ and $A\tau\in\mathbb{L}$ for any period of $\wp(z)$.

Remark 1. It is known that if $h(z)$ satisfies one of (1)-(3) in Theorem 1, then the order $\rho(h(z))\geq 1$ and hyper-order $\rho_{2}(f)\geq 1$. (The fact can be found in [14]). So, if $\rho_{2}(f)<1$, then (1.4) does not admit nonconstant meromorphic solutions. This is the conclusion of Theorem B. Below, we offer example to show that there exist entire function $h(z)$ with arbitrary order $\rho(h(z))(\geq 1)$ satisfying one of (1)-(3).

Example 2. In [39, Theorem 3], Ozawa derived that there exists periodic entire function of arbitrary order $\sigma(\geq 1)$. So, there exists an entire function $g(z)$ such that $\rho(g(z))=\sigma$ and $g(z+c)=g(z)$. Set $h(z)=e^{az}g(z)+b$, where $a$, $b$ are constants with $e^{ac}=A$ and $(1-A)b=\theta_{1}+\tau_{1}$ or $\theta_{2}+\tau_{2}$ or $\tau_{3}$ with $A^{3}=-1$. Then, $\rho(h(z))=\rho(g(z))=\sigma$ and a calculation yields that

Thus, $h(z)$ satisfies one of (1)-(3) in Theorem 1.

By Theorem 1 and Theorem C, we can get the following result.

(a). $f(z)=de^{\frac{\alpha z+\beta}{3}}$, where $d$ is a nonzero constant and $d^{3}(1+e^{\alpha c})=1$;

(b). $f(z)=e^{\frac{\alpha z+\beta}{3}}\frac{1}{2}\left\{1+\frac{\wp^{\prime}(h(z))}{\sqrt{3}}\right\}/\wp(h(z))$, where $e^{\alpha c}=1$ and $h(z)$ is a nonconstant entire function satisfies one of (1)-(3) in Theorem 1.

With Proposition 2, we derive a theorem as follows.

Remark 2. We point out that equation (1.9) may have nonconstant meromorphic functions if $\rho(g(z))\geq 1$, as shown by the following examples.

Example 3. Suppose that $m\neq n$, $g(z)=e^{\alpha z+\beta}+a$ with constants $\alpha(\neq 0),~{}\beta,~{}a$. Consider $f(z)=Ae^{\frac{g(z)}{n}}$ with $e^{\alpha c}=\frac{n}{m}$ and $(1-\frac{n}{m})a=nLn\frac{B}{A}$, where $A,~{}B$ are nonzero constants with $A^{n}+B^{m}=1$. By a calculation, one gets $g(z+c)=\frac{ng(z)}{m}+nLn\frac{B}{A}$ and

Obviously, $\rho(g(z))=1$.

Example 4. Suppose that $m=n$, and $h$ is any nonconstant entire periodic function with period $c$. Let $g(z)=h(z)+az$ with $ac=nLn\frac{B}{A}$, where $A,~{}B$ are nonzero constants with $A^{n}+B^{n}=1$. Then, we have $g(z+c)=g(z)+nLn\frac{B}{A}$, and $f(z)=Ae^{\frac{g(z)}{n}}$ is a nonconstant meromorphic solution of

Obviously, $\rho(g(z))\geq 1$.

Remark 3. From Theorem 3, we see that the equation (1.9) do not admit meromorphic solutions with poles. It is pointed out that the same argument in Theorems 3 can deal with the Fermat-type difference equations $f^{m}(z)+f^{n}(z+c)=e^{g(z)}$ with $n(\geq 2),~{}m(\geq 3)$ and $(m,n)\neq(3,3)$, we omit the details here.

The following corollary follows immediately from Theorem 3.

From Theorem 3 and Corollary 1, one can easily get the following result.

Finally, as applications of Propositions 1 and 2, we consider meromorphic solutions of Fermat-type $q$-difference functional equations. As early as 1952, Valiron in [41] showed that the non-autonomous Schröder $q$-difference equation

where $R(z,f(z))$ is rational in both arguments, admits a one parameter family of meromorphic solutions, provided that $q\in\mathbb{C}$ is chosen appropriately. Later, Gundersen et al [18] proved that if $|q|>1$ and the $q$-difference equation (1.12) admits a meromorphic solution of order zero, then (1.12) reduces to a $q$-difference Riccati equation, i.e. $\deg_{f}R=1$. Some scholars, such as Bergweiler-Hayman [5], Bergweiler-Ishizaki- Yanagihara [6], Eremenko-Sodin [13], Ishizaki-Yanagihara [24] also made contributions to meromorphic solutions of $q$-difference functional equations. In 2007, Barneet-Halburd-Korhonen-Morgan in [3] derived the $q$-difference analogues of some well-known results in Nevanlinna theory, including the lemma of the logarithmic derivative, the Clunie’s lemma and the second main theorem. With the help of results, meromorphic solutions to $q$-difference functional equations were further studied, see [26, 32].

Observe that in Propositions 1 and 2, $L(z)$ reduce to a linear function, say $qz+c$ with $q\neq 0$. Therefore, we have described meromorphic solutions of Fermat-type $q$-difference functional equations $f^{3}(z)+f^{3}(qz+c)=1$ and $f^{n}(z)+f^{n}(qz+c)=e^{g(z)}$, where $n,~{}m$ satisfy some condition and $g(z)$ is an entire function.

Remark 4. From the above theorems, we see that the case $(1,n),~{}(m,1),~{}(2,2)$ are left. Unfortunately, we cannot deal with these cases and thus leave them for further study.

For the proofs, we will assume that the reader is familiar with basic elements in Nevanlinna theory of meromorphic functions in $\mathbb{C}$ (see e.g. [21, 23, 28, 46]), such as the first and second main theorems, the characteristic function $T(r,f)$, the proximity function $m(r,f)$, the counting function $N(r,f)$ and the reduced counting function $\overline{N}(r,f)$. We also need the following notation.

The lower order of a meromorphic function $f$ is defined as

In this section, we firstly give the proof of Proposition 1.