Mathematicshttps://repository.iitgn.ac.in/handle/123456789/5472018-06-24T20:16:39Z2018-06-24T20:16:39ZPartition implications of a new three parameter q-series identityDixit, AtulMaji, Bibekanandahttps://repository.iitgn.ac.in/handle/123456789/37672018-06-21T11:41:37Z2018-06-01T00:00:00ZPartition implications of a new three parameter q-series identity
Dixit, Atul; Maji, Bibekananda
A generalization of a beautiful q-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of partition-theoretic information. In particular, we use this identity to obtain a generalization of a recent result of Andrews, Garvan and Liang, which itself generalizes the famous result of Fokkink, Fokkink and Wang. This three-parameter identity also leads to several new weighted partition identities as well as a natural proof of a recent result of Garvan. This natural proof gives interesting number-theoretic information along the way. We also obtain a new result consisting of an infinite series involving a special case of Fine's function F(a,b;t), namely, F(0,qn;cqn). For c=1, this gives Andrews' famous identity for spt(n) whereas for c=−1,0 and q, it unravels new relations that the divisor function d(n) has with other partition-theoretic functions such as the largest parts function lpt(n)
2018-06-01T00:00:00ZZeros of combinations of the Riemann Ξ-function and the confluent hypergeometric function on bounded vertical shiftsDixit, AtulKumar, RahulMaji, Bibekanandahttps://repository.iitgn.ac.in/handle/123456789/37492018-06-21T10:38:44Z2018-06-01T00:00:00ZZeros of combinations of the Riemann Ξ-function and the confluent hypergeometric function on bounded vertical shifts
Dixit, Atul; Kumar, Rahul; Maji, Bibekananda
In 1914, Hardy proved that infinitely many non-trivial zeros of the Riemann zeta function lie on the critical line using the transformation formula of the Jacobi theta function. Recently the first author obtained an integral representation involving the Riemann Ξ-function and the confluent hypergeometric function linked to the general theta transformation. Using this result, we show that a series consisting of bounded vertical shifts of a product of the Riemann Ξ-function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line, thereby generalizing a previous result due to the first and the last authors along with Roy and Robles. The latter itself is a generalization of Hardy's theorem.
2018-06-01T00:00:00ZOn the complexity of two dots for narrow boards and few colorsBil�, DavideGual�, LucianoLeucci, StefanoMisra, Neeldharahttps://repository.iitgn.ac.in/handle/123456789/37402018-06-13T12:51:59Z2018-06-13T00:00:00ZOn the complexity of two dots for narrow boards and few colors
Bil�, Davide; Gual�, Luciano; Leucci, Stefano; Misra, Neeldhara
2018-06-13T00:00:00ZSymmetric numerical semigroups formed by concatenation of arithmetic sequencesMehta, RanjanaSaha, JoydipSengupta, Indranathhttps://repository.iitgn.ac.in/handle/123456789/37232018-06-07T12:33:41Z2018-05-01T00:00:00ZSymmetric numerical semigroups formed by concatenation of arithmetic sequences
Mehta, Ranjana; Saha, Joydip; Sengupta, Indranath
Given integer e?4, we have constructed a class of symmetric numerical semigroups of embedding dimension e and proved that the cardinality of a minimal presentation of the semigroup is a bounded function of the embedding dimension e. This generalizes the examples given by J.C. Rosales.
2018-05-01T00:00:00Z