On limiting distributions of Graham, Knuth, Patashnik recurrences
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917928773877760 |
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| author | Hitczenko, Pawel |
| author_facet | Hitczenko, Pawel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_13382 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On limiting distributions of Graham, Knuth, Patashnik recurrences Hitczenko, Pawel Probability Combinatorics 05A16, 60C05, 60F05 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. |
| title | On limiting distributions of Graham, Knuth, Patashnik recurrences |
| topic | Probability Combinatorics 05A16, 60C05, 60F05 |
| url | https://arxiv.org/abs/2502.13382 |