Asymptotic behaviour of solutions to difference equations involving ratios of elementary symmetric polynomials

作者：Berenhaut Kenneth S^*; Jones Austin H

来源：Journal of Difference Equations and Applications, 2012, 18(6): 963-972.

DOI：10.1080/10236198.2010.535526

• difference equations
• asymptotic behaviour
• symmetric functions
• elementary symmetric polynomials
• Newton%26apos
• s inequalities
• periodicity

This paper studies the behaviour of positive solutions of the recursive equation y(n) = (e(i,k)/e(j,k))(y(n-t1), y(n-t2), ... , y(n-tk)), 0 %26lt;= i, j %26lt;= k, where e(m,k) is the mth elementary symmetric polynomial on k variables, t(l) %26gt;= 1 for 1 %26lt;= l %26lt;= k, gcd(t(1), t(2), ... , t(k)) = 1 and y(-s), y(-s+1), ... , y(-1) is an element of R+, with s = max{t(1), t(2), ... , t(k)}. A variant of Newton%26apos;s inequalities is employed. Included among the results is a generalization of a particular case of Theorem 4.11 in E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman %26 Hall/CRC Press, Boca Raton, 2004.

• 出版日期2012