On a class of solvable difference equations generalizing an iteration process for calculating reciprocals

[摘要] The well-known first-order nonlinear difference equation$$ y_{n+1}=2y_{n}-xy_{n}^{2}, \quad n\in {\mathbb {N}}_{0}, $$ naturally appeared in the problem of computing the reciprocal value of a given nonzero real number x. One of the interesting features of the difference equation is that it is solvable in closed form. We show that there is a class of theoretically solvable higher-order nonlinear difference equations that include the equation. We also show that some of these equations are also practically solvable.

[学科分类] 航空航天科学

[关键词] Difference equation;Solvable equation;Theoretical solvability;Practical solvability;Closed-form formula