Ramanujan’s five entries, weighted partition identities and divisor generating q-series with applications to probability theory and random graphs

Abstract

Ramanujan recorded many q-series identities in his notebooks and lost notebook. At the end of his second notebook, he mentioned five q-series identities. In 2021, Dixit and Maji obtained a q-series identity that enabled them to derive three of five Ramanujan’s q-series identities. Later, a unified generalization of these five q-series identities was obtained by Bhoria, Eyyunni and Maji and a finite analogue of this generalization was subsequently established by Dixit and Patel, yielding finite analogues of all five identities of Ramanujan. One of the primary objectives of this thesis is to develop a one variable generalization of the aforementioned identity of Bhoria et. al., together with its finite analogue, thereby extending the work of Dixit and Patel. As a consequence, we derive one variable generalizations of each of Ramanujan’s five identities along with their corresponding finite analogues.