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Control of cancellations that restrain the growth of a binomial recursion

Aspenberg, Magnus LU and Perez, Rodrigo (2015) In Journal of Geometric Analysis 25(3). p.1666-1700
Abstract
We study a recursion that generates real sequences depending on a parameter x. Given a negative x the growth of the sequence is very difficult to estimate due to canceling terms. We reduce the study of the recursion to a problem about a family of integral operators, and prove that for every parameter value except -1, the growth of the sequence is factorial. In the combinatorial part of the proof we show that when x=-1 the resulting recurrence yields the sequence of alternating Catalan numbers, and thus has exponential growth. We expect our methods to be useful in a variety of similar situations.
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Catalan numbers, Factorial growth, Integral operators
in
Journal of Geometric Analysis
volume
25
issue
3
pages
1666 - 1700
publisher
Springer
external identifiers
  • wos:000356515800014
  • scopus:84931560812
ISSN
1559-002X
DOI
10.1007/s12220-014-9489-y
language
English
LU publication?
yes
id
005c5b1c-2e31-4e43-ac43-24dec369d7c9 (old id 4251395)
date added to LUP
2014-05-16 16:39:56
date last changed
2017-03-14 13:41:42
@article{005c5b1c-2e31-4e43-ac43-24dec369d7c9,
  abstract     = {We study a recursion that generates real sequences depending on a parameter x. Given a negative x the growth of the sequence is very difficult to estimate due to canceling terms. We reduce the study of the recursion to a problem about a family of integral operators, and prove that for every parameter value except -1, the growth of the sequence is factorial. In the combinatorial part of the proof we show that when x=-1 the resulting recurrence yields the sequence of alternating Catalan numbers, and thus has exponential growth. We expect our methods to be useful in a variety of similar situations.},
  author       = {Aspenberg, Magnus and Perez, Rodrigo},
  issn         = {1559-002X},
  keyword      = {Catalan numbers,Factorial growth,Integral operators},
  language     = {eng},
  number       = {3},
  pages        = {1666--1700},
  publisher    = {Springer},
  series       = {Journal of Geometric Analysis},
  title        = {Control of cancellations that restrain the growth of a binomial recursion},
  url          = {http://dx.doi.org/10.1007/s12220-014-9489-y},
  volume       = {25},
  year         = {2015},
}