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Meromorphic Solutions on the First Order Non-Linear Difference Equations
Analysis Mathematica ( IF 0.7 ) Pub Date : 2020-05-15 , DOI: 10.1007/s10476-020-0032-z
J. Li , K. Liu

Steinmetz [16] considered the first order non-linear differential equations $$C(z, f)(f^\prime)^2+B(z, f)f^\prime+A(z, f)=0,$$ C ( z , f ) ( f ′ ) 2 + B ( z , f ) f ′ + A ( z , f ) = 0 , where A ( z , f ), B ( z , f ), C ( z , f ) are polynomials in f with rational coefficients in z and pointed out that the above equation must reduce into some certain types when it admits transcendental meromorphic solutions. In this paper, we will consider its difference version $$C(z, f)f(z+c)^2+B(z, f)f(z+c)+A(z,f)=0.$$ C ( z , f ) f ( z + c ) 2 + B ( z , f ) f ( z + c ) + A ( z , f ) = 0. We explore the conditions when the above difference equation admits transcendental meromorphic (entire) solutions. In addition, the difference equations which are similar to Fermat difference equations also be selected out and considered.

中文翻译:

一阶非线性差分方程的亚纯解

Steinmetz [16] 考虑了一阶非线性微分方程 $$C(z, f)(f^\prime)^2+B(z, f)f^\prime+A(z, f)=0, $$ C ( z , f ) ( f ′ ) 2 + B ( z , f ) f ′ + A ( z , f ) = 0 , 其中 A ( z , f ), B ( z , f ), C ( z , f ) 是 f 中的多项式,z 中的系数为有理数,并指出上述方程在允许超越亚纯解时必须归约为某些类型。在本文中,我们将考虑其差异版本 $$C(z, f)f(z+c)^2+B(z, f)f(z+c)+A(z,f)=0.$ $ C ( z , f ) f ( z + c ) 2 + B ( z , f ) f ( z + c ) + A ( z , f ) = 0. 我们探讨了上述差分方程承认超越亚纯 (整个)解决方案。此外,与费马差分方程相似的差分方程也被选择和考虑。
更新日期:2020-05-15
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