# Correction to: Solutions of Complex Fermat-Type Partial Difference and Differential-Difference Equations

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## 1 Correction to: Mediterr. J. Math. (2018) 15:227 https://doi.org/10.1007/s00009-018-1274-x

**Abstract**. We give a correction to Theorem 1.2 in a previous paper [Mediterr. J. Math. (2018) 15:227]. Two examples are given to explain the corrected conclusion.

**Mathematics Subject Classification**. Primary 39A45 Secondary 32H30 39A14 35A20

**Keywords**. Several complex variables meromorphic functions fermat-type equations Nevanlinna theory partial differential-difference equations

## 2 Introduction and main result

Recently, the present authors originally considered solutions of complex partial differential-difference equations of the Fermat type by making use of Nevanlinna theory. Unfortunately, there was an error in the proof of [1, Theorem 1.2] (that is lines -1 to -3 on the Page 11), and thus its conclusion was stated wrong. Here we correct it as follows.s

### Theorem 1.1

*A*,

*B*are constants on \(\mathbb {C}\) satisfying \(A^2=1\) and \(Ae^{i(A c_{1}+Bc_{2})}=1,\) and \(H(z_{2})\) is a polynomial in one variable \(z_{2}\) such that \(H(z_{2})\equiv H(z_{2}+c_{2}).\) In the special case whenever \(c_{2}\ne 0,\) we have \(f(z_{1}, z_{2})=\sin \left( Az_{1}+Bz_{2}+\text{ Constant }\right) .\)

We show the details of the proof as follows.

### Proof

*f*is a transcendental entire solution with finite order of equation (1), then

*p*is a nonconstant entire function on \(\mathbb {C}^{2},\) which gives

*p*should be a polynomial function on \(\mathbb {C}^{2}.\) Hence,

*p*is a nonconstant polynomial on \(\mathbb {C}^{2}.\) From these equations above, we get from [1, Lemma 3.2] that

*p*is a nonconstant polynomial, we see that \(ip(z_{1}, z_{2})+ip(z_{1}+c_{1}, z_{2}+c_{2})\) can not be a constant. This implies that both \(e^{i2p(z_{1}+c_{1}, z_{2}+c_{2})}\) and \(e^{ip(z_{1}, z_{2})+ip(z_{1}+c_{1}, z_{2}+c_{2})}\) must be nonconstant and transcendental on \(\mathbb {C}^{2},\) and that

*A*is a nonzero constant in \(\mathbb {C}.\) Submitting (3) into (2) gives

*B*in the original proof in [1], and thus the following is different from the original proof). Since \(p(z_{1}, z_{2})-p(z_{1}+c_{1}, z_{2}+c_{2})=-i Ln A,\) we get that

We give two examples to explain the conclusion of the theorem.

### Example 1.2

### Example 1.3

If there are no differences, that is \(c=(0,0),\) then Theorem 1.1 implies the following corollary.

### Corollary 1.4

## Notes

## Reference

- 1.Xu, L., Cao, T.B.: Solutions of Complex Fermat-Type Partial Difference and Differential-Difference Equations. Mediterr. J. Math.
**15**, 227 (2018). https://doi.org/10.1007/s00009-018-1274-x MathSciNetCrossRefzbMATHGoogle Scholar