Separating Solution of a Quadratic Recurrent Equation

Abstract

In this paper we consider the recurrent equation $${\Lambda }_{p+1}=\frac{1}{p}\sum _{q=1}^{p}f\left(\frac{q}{p+1}\right){\Lambda }_q{\Lambda }_{p+1-q}$$ for $p\ge 1$ with $f\in C[0,1]$ and ${\Lambda }_1=y>0$ given. We give conditions on $f$ that guarantee the existence of $y^{(0)}$ such that the sequence ${\Lambda }_p$ with ${\Lambda }_1=y^{(0)}$ tends to a finite positive limit as $p\to \infty$.

13 pages, 6 figures, submitted to J. Stat. Phys

References in corpus (1)

• On a nonlinear recurrent relation