2020/11/16 by Subhash Chand Bhoria, Pramod Eyyunni, Bhoria, Subhash Chand +3
Mathematics · #05A17 #11P81 #11P84 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2011.07767
openalex publication_date 2020/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Ramanujan recorded five interesting q-series identities in a section that is\nnot as systematically arranged as the other chapters of his second notebook.\nThese five identities do not seem to have acquired enough attention. Recently,\nDixit and the third author found a one-variable generalization of one of the\naforementioned five identities. From their generalized identity, they were able\nto derive the last three of these q-series identities, but didn't establish the\nfirst two. In the present article, we derive a one-variable generalization of\nthe main identity of Dixit and the third author from which we successfully\ndeduce all the five q-series identities of Ramanujan. In addition to this, we\nalso establish a few interesting weighted partition identities from our\ngeneralized identity. In the mid 1980's, Bressoud and Subbarao found an\ninteresting identity connecting the generalized divisor function with a\nweighted partition function, which they proved by means of a purely\ncombinatorial argument. Quite surprisingly, we found an analytic proof for a\ngeneralization of the identity of Bressoud and Subbarao, starting from the\nfourth identity of the aforementioned five q-series identities of Ramanujan.\n