scholarly journals Even–odd partition identities of Rogers–Ramanujan type

2021 ◽  
Author(s):  
Pooneh Afsharijoo
Keyword(s):  
Partition Identities

scholarly journals Proofs of some partition identities conjectured by Kanade and Russell

2021 ◽  
Author(s):  
Hjalmar Rosengren
Keyword(s):  
Lie Algebra ◽  
Partition Identities ◽  
Partition Identity ◽  
Affine Lie Algebra ◽  
Level 2

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.

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

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.

scholarly journals Partition identities II. The results of Bateman and Erdős

2006 ◽  
Vol 117 (1) ◽  
pp. 160-190
Author(s):  
Jason P. Bell ◽  
Stanley N. Burris
Keyword(s):  
Partition Identities

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

10.37236/1796 ◽  
2004 ◽  
Vol 11 (1) ◽  
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.

scholarly journals A Family of Theta-Function Identities Based upon Combinatorial Partition Identities Related to Jacobi’s Triple-Product Identity

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 918
Author(s):  
Hari Mohan Srivastava ◽  
Rekha Srivastava ◽  
Mahendra Pal Chaudhary ◽  
Salah Uddin
Keyword(s):  
Theta Function ◽  
Subject Matter ◽  
Open Problem ◽  
Triple Product ◽  
Partition Identities ◽  
Product Identity ◽  
Recent Developments ◽  
The Subject ◽  
Theta Function Identities ◽  
R Functions

The authors establish a set of six new theta-function identities involving multivariable R-functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper, we consider and relate the multivariable R-functions to several interesting q-identities such as (for example) a number of q-product identities and Jacobi’s celebrated triple-product identity. Various recent developments on the subject-matter of this article as well as some of its potential application areas are also briefly indicated. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities and present a presumably open problem.

scholarly journals Proofs and reductions of various conjectured partition identities of Kanade and Russell

2020 ◽  
Vol 2020 (766) ◽  
pp. 109-135 ◽  
Author(s):  
Kathrin Bringmann ◽  
Chris Jennings-Shaffer ◽  
Karl Mahlburg
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Partition Identities ◽  
Symmetric Identities ◽  
Level 2

AbstractWe prove seven of the Rogers–Ramanujan-type identities modulo 12 that were conjectured by Kanade and Russell. Included among these seven are the two original modulo 12 identities, in which the products have asymmetric congruence conditions, as well as the three symmetric identities related to the principally specialized characters of certain level 2 modules of {A_{9}^{(2)}}. We also give reductions of four other conjectures in terms of single-sum basic hypergeometric series.

BIJECTIVE PROOFS OF A THEOREM OF FINE AND RELATED PARTITION IDENTITIES

2009 ◽  
Vol 05 (02) ◽  
pp. 219-228 ◽  
Author(s):  
AE JA YEE
Keyword(s):  
Partition Identities

In this paper, we prove a theorem of Fine bijectively. Stacks with summits and gradual stacks with summits are also discussed.

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