Journal of Combinatorial Theory Series A· 2026Q1
On limiting distributions of Graham, Knuth, Patashnik recurrences
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- Q1SCImago
- 2026year
Short summary
This paper provides a complete description of the limiting distributions for a class of generalized recurrences, specifically when the parameter α' is zero and the other five parameters are non-negative.
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Key points
- Analyzes the limiting distributions of random variables associated with generalized recurrences.
- Considers recurrences of the form |n k|=(αn+βk+γ)|n-1 k|+(α'n+β'k+γ')|n-1 k-1|+I_{n=k=0}.
- Provides a complete description of limiting behavior when α'=0 and other five parameters are non-negative.
AI-generated from the title and abstract; the full text is not read.
Abstract
Graham, Knuth and Patashnik in their book Concrete Mathematics called for development of a general theory of the solutions of recurrences defined by $$\left|{ n\atop k}\right|=(αn+βk+γ)\left|{n-1\atop k}\right|+(α' n+β' k+γ')\left|{n-1\atop k-1}\right|+I_{n=k=0}$$ for $0\le k\le n$ and six parameters $α,β,γ,α'β',γ'$. 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 $α'=0$ and the remaining five parameters are non--negative.
The authors' abstract, as published at the source. Journal of Combinatorial Theory Series A, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics