Solutions of Complex Fermat-Type Partial Difference and Differential-Difference Equations

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

“…where g(z 2 ) should be a polynomial function in one variable z 2 (Note that the present authors made a mistake of p(z 1 , z 2 ) = Az 1 + B with a constant B in the original proof in [1], and thus the following is different from the original proof). Since…”

“…Furthermore, it follows immediately from [1,Lemma 3.3] for any variable z j (j ∈ 1, 2) that p should be a polynomial function on C 2 . Hence, p is a nonconstant polynomial on C 2 .…”

“…Hence, p is a nonconstant polynomial on C 2 . From these equations above, we get from [1,Lemma 3.2] that z1+c1,z2+c2) + e −ip(z1+c1,z2+c2) 2 z1+c1,z2+c2)−ip(z1,z2) −e i2p(z1+c1,z2+c2) = 1.…”

Entire solutions for several second-order partial differential-difference equations of Fermat type with two complex variables

“…, where f is an entire function in C satisfying f (c 1 z 1 + c 2 z 2 ) = ±e 1 2 g(z) , c 1 and c 2 are two constants satisfying c 2 1 + c 2 2 = 1, and φ 1 and φ 2 are entire functions in C satisfying φ 1 (z 1 + iz 2 )φ 2 (z 1iz 2 ) = 1 4 e g (z) .…”

“…Very recently, Xu and Cao [1,2,29] investigated the existence of solutions for some Fermat-type partial differential-difference equations with several variables by using the difference logarithmic derivative lemma of several complex variables and obtained the following theorem (see [29][30][31]).…”