Weighted partition identities and applications

AbstractKanade and Russell conjectured several Rogers–Ramanujan-type partition identities, some of which are related to level 2 characters of the affine Lie algebra $$A_9^{(2)}$$ A 9 ( 2 ) . Many of these conjectures have been proved by Bringmann, Jennings-Shaffer and Mahlburg. We give new proofs of five conjectures first proved by those authors, as well as four others that have been open until now. Our proofs for the new cases use quadratic transformations for Askey–Wilson and Rogers polynomials. We also obtain some related results, including a partition identity conjectured by Capparelli and first proved by Andrews.

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New proof to Somos’s Dedekind eta-function identities of level 10

Soft Computing ◽

10.1007/s00500-021-05846-7 ◽

2021 ◽

Author(s):

B. R. Srivatsa Kumar ◽

Shruthi

Keyword(s):

Dedekind Eta Function ◽

Partition Identities

AbstractMichael Somos used PARI/GP script to generate several Dedekind eta-function identities by using computer. In the present work, we prove two new Dedekind eta-function identities of level 10 discovered by Somos in two different methods. Also during this process, we give an alternate method to Somos’s Dedekind eta-function identities of level 10 proved by B. R. Srivatsa Kumar and D. Anu Radha. As an application of this, we establish colored partition identities.

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Partition identities with mixed mock modular forms

Journal of Number Theory ◽

10.1016/j.jnt.2015.06.022 ◽

2016 ◽

Vol 158 ◽

pp. 356-364 ◽

Cited By ~ 1

Author(s):

George E. Andrews ◽

Stephen Hill

Keyword(s):

Modular Forms ◽

Partition Identities ◽

Mock Modular Forms ◽

Mixed Mock Modular Forms

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Partition identities II. The results of Bateman and Erdős

Journal of Number Theory ◽

10.1016/j.jnt.2005.06.011 ◽

2006 ◽

Vol 117 (1) ◽

pp. 160-190

Author(s):

Jason P. Bell ◽

Stanley N. Burris

Keyword(s):

Partition Identities

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Bijections and Congruences for Generalizations of Partition Identities of Euler and Guy

The Electronic Journal of Combinatorics ◽

10.37236/1796 ◽

2004 ◽

Vol 11 (1) ◽

Cited By ~ 4

Author(s):

James A. Sellers ◽

Andrew V. Sills ◽

Gary L. Mullen

Keyword(s):

Famous Theorem ◽

Partition Identities ◽

Product Identity ◽

Powers Of 2 ◽

Pentagonal Number Theorem ◽

Series Product

In 1958, Richard Guy proved that the number of partitions of $n$ into odd parts greater than one equals the number of partitions of $n$ into distinct parts with no powers of 2 allowed, which is closely related to Euler's famous theorem that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We consider extensions of Guy's result, which naturally lead to a new algorithm for producing bijections between various equivalent partition ideals of order 1, as well as to two new infinite families of parity results which follow from Euler's Pentagonal Number Theorem and a well-known series-product identity of Jacobi.

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