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AN ENTIRE FUNCTION DEFINED BY A NONLINEAR RECURRENCE RELATION

Published online by Cambridge University Press:  24 March 2003

A. N. W. HONE ,
N. JOSHI  and
A. V. KITAEV
A. N. W. HONE
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide 5005, Australia Current address: Institute of Mathematics and Statistics, University of Kent, Canterbury, Kent CT2 7NF anwh@ukc.ac.uk
N. JOSHI
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide 5005, Australia Current address: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australianalini@maths.usyd.edu.au
A. V. KITAEV
Affiliation:
Department of Pure Mathematics, University of Adelaide, Adelaide 5005, Australia Current address: Steklov Mathematical Institute, Fontanka 27, 191011 Russiakitaev@pdmi.ras.ru
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Abstract

For the nonlinear recurrence relation \[ \alpha_2 = \alpha_1(1 - \alpha_1), \quad \alpha_{k+1} = k^2 \alpha_k + \sum\limits_{m=2}^{k-1} \alpha_m \alpha_{k+1-m}, \quad k\geqslant 2, \] it is proved that the limit \[ p_{\infty}(\alpha_1) = \lim_{k \rightarrow \infty} \alpha_k /[(k-1)!]^2 \] exists and defines an entire function of $\alpha_2 = \alpha_1(1-\alpha_1)$ .

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 66 , Issue 2 , October 2002 , pp. 377 - 387
Copyright
The London Mathematical Society, 2002

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