Proofs and reductions of various conjectured partition identities of Kanade and Russell
TLDR
In this article, the authors proved seven of the Rogers-Ramanujan type identities modulo $12$ that were conjectured by Kanade and Russell and gave reductions of four other conjectures in terms of single-sum basic hypergeometric series.Abstract:
We 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.read more
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Andrews-Gordon Type Series for Kanade-Russell Conjectures
TL;DR: In this paper, the Andrews-Gordon type evidently positive series conjectures were constructed as generating functions of partitions satisfying certain difference conditions in six conjectures by Kanade and Russell, without claiming new partition identities.
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Proofs of some partition identities conjectured by Kanade and Russell
TL;DR: In this article, the authors give new proofs of five conjectures first proved by Kanade and Russell, as well as four others that have been open until now, using quadratic transformations for Askey-Wilson and Rogers polynomials.
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A proof of conjectured partition identities of Nandi
TL;DR: The authors generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers-Ramanujan type identities of modulo 14 that were posed by Nandi through vertex operator theoretic construction of the level 4 standard modules of the affine Lie algebra.
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qFunctions – A Mathematica package for q-series and partition theory applications
Jakob Ablinger,Ali Kemal Uncu +1 more
TL;DR: The qfunctions Mathematica package as discussed by the authors includes both experimental and symbolic tools for q-series and partition theory applications, including guessers for qdifference equations and recurrences for given q-difference and fitting/finding explicit expressions for sequences of polynomials.
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Linked partition ideals and Kanade–Russell conjectures
Shane Chern,Zhitai Li +1 more
TL;DR: In this article, a method of proving generating function identities for partitions from linked partition ideals is presented. But the method is built on a conjecture by George Andrews and that those generating functions satisfy some q -difference equations.
References
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Book
Principles of mathematical analysis
TL;DR: The real and complex number system as discussed by the authors is a real number system where the real number is defined by a real function and the complex number is represented by a complex field of functions.
Book
Infinite Dimensional Lie Algebras
TL;DR: The invariant bilinear form and the generalized casimir operator are integral representations of Kac-Moody algebras and the weyl group as mentioned in this paper, as well as a classification of generalized cartan matrices.
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Basic Hypergeometric Series
George Gasper,Mizan Rahman +1 more
TL;DR: In this article, the Askey-Wilson q-beta integral and some associated formulas were used to generate bilinear generating functions for basic orthogonal polynomials.