Solutions of Fermat-Type Partial Differential–Difference Equations in $$\pmb {{\mathbb {C}}^n}$$

Abstract

The equation f n + g n = 1, n ∈ N can be regarded as the Fermat Diophantine equation over the function field. In this paper we study the characterization of entire solutions of some system of Fermat type functional equations by taking e g1(z) and e g2(z) in the right hand side of each equation, where g 1 (z) and g 2 (z) are polynomials in C n . Our results extend and generalize some recent results. Moreover, some examples have been exhibited to show that our results are precise to some extent.

Transcendental entire solutions of quadratic functional equations

“…We conclude some remark corresponding to Theorem 3.2. (i) We note that Theorem 3.2 improves the results of[27, Theorem 1.3] and[18, Theorem 1.3]. (ii) If n = 1, P (z) = 1, Q(z) = 1, a = 1, b = 1 and g(z) = 2kπi, k ∈ Z, thenwe obtain the conclusion of[27, Theorem 1.3].…”

“…Thus, our result exhibits a higher level of improvement compared to previous results. (iii) In particular, if n = 2, a = 1, b = 1 and g(z) = 2kπi, k ∈ Z, one can obtain the conclusions of[18, Theorem 1.3] in C 2 . Accordingly, our result can be regarded as a more refined and improved version in comparison to the existing ones.…”

“…We refer to the articles [40,42,43] and references therein for some recent development on the topic. In the past decade, with the establishment of difference analogues of Nevanlinna theory (see e.g., [4,8,16]), many researchers studied the solutions of Fermat type difference and also Fermat-type differential-difference equations (see [18,28,34]). Here, an equation is called differential-difference equation, if it includes derivatives, shifts or differences of f (z), which can be called DDE for short (see [31]).…”

“…By employing Nevanlinna theory and the method of complex analysis, there were a number of literature focusing on the solutions of some PDEs and their many variants, readers can refer to [6,15,21,25,38]. The solutions of Fermat-type P DEs were investigated by [18,23,36]. Most noticeably, in 1995, Khavinson [21] derived that any entire solution of the partial differential equation in C 2 ,…”

Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in $$ {\mathbb {C}}^2 $$

“…In 1995, Khavinson pointed out that in C 2 , the entire solution of the Fermat type partial differential equations f 2 z 1 + f 2 z 2 = 1 must necessarily be linear. After the development of the difference Nevanlinna theory in several complex variables, specially the difference version of logarithmic derivative lemma (see [1][2][3]20]), many Researchers started studying the existence and the precise form of entire and meromorphic solutions of different variants of Fermat type difference and partial differential difference equations, and obtained very remarkable and interesting results (see [15,16,34,38,[42][43][44]46]).…”