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Preprint
On limiting distributions of Graham, Knuth, Patashnik recurrences
18 Feb 2025
Abstract
Graham, Knuth and Patashnik in their book Concrete Mathematics called for
development of a general theory of the solutions of recurrences defined by
$$\left|{ n\atop k}\right|=(\alpha n+\beta k+\gamma)\left|{n-1\atop
k}\right|+(\alpha' n+\beta' k+\gamma')\left|{n-1\atop k-1}\right|+I_{n=k=0}$$
for $0\le k\le n$ and six parameters
$\alpha,\beta,\gamma,\alpha'\beta',\gamma'$. Since then, a number of authors
investigated various properties of the solutions of these recurrences. In this
note we consider a probabilistic aspect, namely we consider the limiting
distributions of sequences of integer valued random variables naturally
associated with the solutions of such recurrences. We will give a complete
description of the limiting behavior when $\alpha'=0$ and the remaining five
parameters are non--negative.
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Details
- Title
- On limiting distributions of Graham, Knuth, Patashnik recurrences
- Creators
- Pawel Hitczenko
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022029670304721