A divisor generating q-series identity and its applications to probability theory and random graphs
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2024
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| _version_ | 1866916233877651456 |
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| author | Agarwal, Archit Bhoria, Subhash Chand Eyyunni, Pramod Maji, Bibekananda |
| author_facet | Agarwal, Archit Bhoria, Subhash Chand Eyyunni, Pramod Maji, Bibekananda |
| contents | In I981, Uchimura studied a divisor generating $q$-series that has applications in probability theory and in the analysis of data structures, called heaps. Mainly, he proved the following identity. For $|q|<1$, \begin{equation*} \sum_{n=1}^\infty n q^n (q^{n+1})_\infty =\sum_{n=1}^{\infty} \frac{(-1)^{n-1} q^{\frac{n(n+1)}{2} } }{(1-q^n) ( q)_n } = \sum_{n=1}^{\infty} \frac{ q^n }{1-q^n}. \end{equation*} Over the years, this identity has been generalized by many mathematicians in different directions. Uchimura himself in 1987, Dilcher (1995), Andrews-Crippa-Simon (1997), and recently Gupta-Kumar (2021) found a generalization of the aforementioned identity. Any generalization of the right most expression of the above identity, we name as divisor-type sum, whereas a generalization of the middle expression we say Ramanujan-type sum, and any generalization of the left most expression we refer it as Uchimura-type sum. Quite surprisingly, Simon, Crippa and Collenberg (1993) showed that the same divisor generating function has a connection with random acyclic digraphs. One of the main themes of this paper is to study these different generalizations and present a unified theory. We also discuss applications of these generalized identities in probability theory for the analysis of heaps and random acyclic digraphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_01877 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A divisor generating q-series identity and its applications to probability theory and random graphs Agarwal, Archit Bhoria, Subhash Chand Eyyunni, Pramod Maji, Bibekananda Number Theory Combinatorics Probability Primary 11P84, 33D15, Secondary 05C80, 60F99 In I981, Uchimura studied a divisor generating $q$-series that has applications in probability theory and in the analysis of data structures, called heaps. Mainly, he proved the following identity. For $|q|<1$, \begin{equation*} \sum_{n=1}^\infty n q^n (q^{n+1})_\infty =\sum_{n=1}^{\infty} \frac{(-1)^{n-1} q^{\frac{n(n+1)}{2} } }{(1-q^n) ( q)_n } = \sum_{n=1}^{\infty} \frac{ q^n }{1-q^n}. \end{equation*} Over the years, this identity has been generalized by many mathematicians in different directions. Uchimura himself in 1987, Dilcher (1995), Andrews-Crippa-Simon (1997), and recently Gupta-Kumar (2021) found a generalization of the aforementioned identity. Any generalization of the right most expression of the above identity, we name as divisor-type sum, whereas a generalization of the middle expression we say Ramanujan-type sum, and any generalization of the left most expression we refer it as Uchimura-type sum. Quite surprisingly, Simon, Crippa and Collenberg (1993) showed that the same divisor generating function has a connection with random acyclic digraphs. One of the main themes of this paper is to study these different generalizations and present a unified theory. We also discuss applications of these generalized identities in probability theory for the analysis of heaps and random acyclic digraphs. |
| title | A divisor generating q-series identity and its applications to probability theory and random graphs |
| topic | Number Theory Combinatorics Probability Primary 11P84, 33D15, Secondary 05C80, 60F99 |
| url | https://arxiv.org/abs/2405.01877 |