Some identities for the partition function

Goswami, A; Jha, A K; Singh, A K

dc.contributor.author	Goswami, A
dc.contributor.author	Jha, A K
dc.contributor.author	Singh, A K
dc.date.accessioned	2022-01-18T11:43:53Z
dc.date.available	2022-01-18T11:43:53Z
dc.date.issued	2021-11-25
dc.identifier.issn	0022247X
dc.identifier.other	10.1016/j.jmaa.2021.125864
dc.identifier.uri	http://localhost:8080/xmlui/handle/123456789/1825
dc.description	This work was done when the first author was a postdoc at Research Institute for Symbolic Computation (RISC), Austria. He was supported by grant SFB F50-06 of the Austrian Science Fund (FWF). The authors thank the anonymous referee for the comments and feedback. The first author thanks Ralf Hemmecke for explaining how the function GroebnerBasis() in Mathematica works. The authors thank Madeline Locus Dawsey, Atul Dixit, Frank Garvan, Ben Kane, Robert Osburn and Peter Paule for their feedback. Our computation have been done in SageMath [34] and Mathematica [	en_US
dc.description.abstract	In his unpublished manuscript on the partition and tau functions, Ramanujan obtained several striking congruences for the partition function p(n), the number of unrestricted partitions of n. The most notable of them are p(5n+4)≡0(mod5) and p(7n+5)≡0(mod7) which holds for all positive integers n. More surprisingly, Ramanujan obtained certain identities between q-series from which the above congruences follow as consequences. In this paper, we adopt Ramanujan's approach and prove an identity which witnesses another famous Ramanujan congruence, namely, p(11n+6)≡0(mod11) and also establish some new identities for the generating functions for p(17n+5),p(19n+7) and p(23n+1). We also find explicit evaluations for Fp(q) in the cases p=17,19,23 where Fp is the function appearing in Ramanujan's circular summation formula.	en_US
dc.description.sponsorship	Research Institute for Symbolic Computation	en_US
dc.language.iso	en_US	en_US
dc.publisher	Academic Press Inc.	en_US
dc.relation.ispartofseries	Journal of Mathematical Analysis and Applications;508,1
dc.subject	Modular forms	en_US
dc.subject	Partition function	en_US
dc.subject	Ramanujan's circular summation	en_US
dc.subject	Ramanujan's congruences	en_US
dc.subject	Witness identity	en_US
dc.title	Some identities for the partition function	en_US
dc.type	Article	en_US