Some Modular Relations Analogues to the Ramanujan’s Forty Identities with Its Applications to Partitions (original) (raw)
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In a manuscript of Ramanujan, published with his Lost Notebook there are forty identities involving the Rogers-Ramanujan functions. In this paper, we establish modular relations involving the Rogers-Ramanujan functions, the Rogers-Ramanujan type functions of order ten and the Rogers-Ramanujan-Slater type functions of order fifteen which are analogues to Ramanujan forty identities. We also give partition theoretic interpretations of our modular relations. 1977, D. Bressoud [12] in his doctoral thesis, proved 15 more from the list of 40. In 1989, A. F. J. Biagioli [9] proved 8 of the remaining 9 identities by invoking the theory of modular forms. Recently, B. C. Berndt et al. [8] have found new proofs for 35 of the forty identities in the spirit of Ramanujan. S. -S. Huang [16] and S. -L. Chen and Huang [13] have established several modular relations for the Göllnitz-Gordan functions by techniques which have been used by Rogers, Watson and Bressoud to prove some of Ramanujan's 40 identities. In 2008, N. D. Baruah, J. Bora and N. Saikia [6] have given alternative proofs some of them by using Schröter's formulas and some simple theta functions identities of Ramanujan. In 2003, H. Hahn [15] has established several modular relations for the septic analogues of the Rogers-Ramanujan functions. In 2007, Baruah and Bora [5] have established several modular relations for the nonic analogues of the Rogers-Ramanujan functions as well as relations that are connected with the Rogers-Ramanujan, Göllnitz-Gordan and septic analogues of Rogers-Ramanujan type functions. In 2007, Baruah and Bora [4] have established several modular relations involving two functions analogues to the Rogers-Ramanujan functions. Some of these relations are connected with Rogers-Ramanujan,
Further analogues of the Rogers-Ramanujan functions with applications to partitions
Combinatorial Number Theory
In this paper, we establish several modular relations involving two functions analogous to the Rogers-Ramanujan functions. These relations are analogous to Ramanujan's famous forty identities for the Rogers-Ramanujan functions. Also, by the notion of colored partitions, we extract partition theoretic interpretations from some of our relations.
Rogers-Ramanujan Functions, Modular Functions, and Computer Algebra
2016
Many generating functions for partitions of numbers are strongly related to modular functions. This article introduces such connections using the Rogers-Ramanujan functions as key players. After exemplifying basic notions of partition theory and modular functions in tutorial manner, relations of modular functions to q-holonomic functions and sequences are discussed. Special emphasis is put on supplementing the ideas presented with concrete computer algebra. Despite intended as a tutorial, owing to the algorithmic focus the presentation might contain aspects of interest also to the expert. One major application concerns an algorithmic derivation of Felix Klein’s classical icosahedral equation.
Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities
The Ramanujan Journal
Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the third author found a one-variable generalization of one of the aforementioned five identities. From their generalized identity, they were able to derive the last three of these q-series identities, but didn't establish the first two. In the present article, we derive a one-variable generalization of the main identity of Dixit and the third author from which we successfully deduce all the five q-series identities of Ramanujan. In addition to this, we also establish a few interesting weighted partition identities from our generalized identity. In the mid 1980's, Bressoud and Subbarao found an interesting identity connecting the generalized divisor function with a weighted partition function, which they proved by means of a purely combinatorial argument. Quite surprisingly, we found an analytic proof for a generalization of the identity of Bressoud and Subbarao, starting from the fourth identity of the aforementioned five q-series identities of Ramanujan.
Formulas for the number of partitions related to the Rogers-Ramanujan identities
Journal of Discrete Mathematical Sciences and Cryptography, 2020
In 2011, Santos, Ribeiro and Mondek have obtained a method, using two-line arrays, to representing partitions. Using this method we provide two formulas for the evaluation of the number of integer partitions of n for classes related to the first and second Rogers-Ramanujan identities, into k £ n parts.