A divisor generating $q$-series and cumulants arising from random graphs
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913951671910400 |
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| author | Agarwal, Archit Bhoria, Subhash Chand Eyyunni, Pramod Maji, Bibekananda Wakhare, Tanay |
| author_facet | Agarwal, Archit Bhoria, Subhash Chand Eyyunni, Pramod Maji, Bibekananda Wakhare, Tanay |
| contents | Uchimura, in 1987, introduced a probability generating function for a random variable $X$ and using properties of this function he discovered an interesting $q$-series identity. He further showed that the $m$-th cumulant with respect to the random variable $X$ is nothing but the generating function for the generalized divisor function $σ_{m-1}(n)$. Simon, Crippa, and Collenberg, in 1993, explored the $G_{n,p}$-model of a random acyclic digraph and defined a random variable $γ_n^{*}(1)$. Quite interestingly, they found links between limit of its mean and the generating function for the divisor function $d(n)$. Later in 1997, Andrews, Crippa and Simon extended these results using $q$-series techniques. They calculated limit of the mean and variance of the random variable $γ_n^{*}(1)$ which correspond to the first and second cumulants. In this paper, we generalize the result of Andrews, Crippa and Simon by calculating limit of the $t$-th cumulant in terms of the generalized divisor function. Furthermore, we also discover limit forms for identities of Uchimura and Dilcher. This provides a fourth side to the Uchimura-Ramanujan-divisor type three way partition identities expounded by the first four authors recently. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15869 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A divisor generating $q$-series and cumulants arising from random graphs Agarwal, Archit Bhoria, Subhash Chand Eyyunni, Pramod Maji, Bibekananda Wakhare, Tanay Number Theory Combinatorics Probability Primary 11P84, 33D15, Secondary 05C80, 60F99 Uchimura, in 1987, introduced a probability generating function for a random variable $X$ and using properties of this function he discovered an interesting $q$-series identity. He further showed that the $m$-th cumulant with respect to the random variable $X$ is nothing but the generating function for the generalized divisor function $σ_{m-1}(n)$. Simon, Crippa, and Collenberg, in 1993, explored the $G_{n,p}$-model of a random acyclic digraph and defined a random variable $γ_n^{*}(1)$. Quite interestingly, they found links between limit of its mean and the generating function for the divisor function $d(n)$. Later in 1997, Andrews, Crippa and Simon extended these results using $q$-series techniques. They calculated limit of the mean and variance of the random variable $γ_n^{*}(1)$ which correspond to the first and second cumulants. In this paper, we generalize the result of Andrews, Crippa and Simon by calculating limit of the $t$-th cumulant in terms of the generalized divisor function. Furthermore, we also discover limit forms for identities of Uchimura and Dilcher. This provides a fourth side to the Uchimura-Ramanujan-divisor type three way partition identities expounded by the first four authors recently. |
| title | A divisor generating $q$-series and cumulants arising from random graphs |
| topic | Number Theory Combinatorics Probability Primary 11P84, 33D15, Secondary 05C80, 60F99 |
| url | https://arxiv.org/abs/2501.15869 |