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A probabilistic proof of the Andrews–Gordon identities

Robin Chapman
2005 Discrete Mathematics  
Recently, Fulman proved the "extreme" cases of the Andrews-Gordon identities using a Markov chain on the non-negative integers.  ...  Here we extend Fulman's methods to prove the Andrews-Gordon identities in full generality.  ...  In [4] Fulman provided a probabilistic proof, involving a Markov chain on the nonnegative integers, of the "extreme" cases of the Andrews-Gordon identities, namely, for i = 1 and i = k.  ... 
doi:10.1016/j.disc.2004.10.018 fatcat:mzyhhkvpvffbremjn3wqdgjs2a

New Proofs of Someq-Summation andq-Transformation Formulas

Xian-Fang Liu, Ya-Qing Bi, Qiu-Ming Luo
2014 The Scientific World Journal  
We obtain an expectation formula and give the probabilistic proofs of some summation and transformation formulas ofq-series based on our expectation formula.  ...  Although these formulas in themselves are not the probability results, the proofs given are based on probabilistic concepts.  ...  Acknowledgments The authors express their sincere gratitude to the referee for many valuable comments and suggestions. The  ... 
doi:10.1155/2014/940358 pmid:24895675 pmcid:PMC4033541 fatcat:cj6sikoyt5hwhn3eozfeyubtli

Page 1248 of Mathematical Reviews Vol. 45, Issue 5 [page]

1973 Mathematical Reviews  
The author remarks that the above method can be i extended to give a proof of the Rogers-Ramanujan- Gordon identities [B. Gordon, Amer. J.  ...  The object of the present paper is to give a proof of the Rogers-Ramanujan identities making use of the fact that two systems of partial q-difference equations are com- 1248 -,m,.  ... 

Page 1724 of Mathematical Reviews Vol. , Issue 98C [page]

1998 Mathematical Reviews  
(English and French summaries) |A probabilistic proof of a result of P. Baird and J. C. Wood] Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 2, 283-291.  ...  The proof is related to the probabilistic interpretation of harmonic morphisms and uses elementary potential analysis of the Brown- ian motion.  ... 

The Rogers-Ramanujan Identities, the Finite General Linear Groups, and the Hall-Littlewood Polynomials [article]

Jason Fulman
1997 arXiv   pre-print
This note connects the Rogers-Ramanujan identities with the finite general linear groups and the Hall-Littlewood polynomials of symmetric function theory.  ...  The Rogers-Ramanujan identities have been studied from the viewpoints of combinatorics, number theory, affine Lie algebras, statistical mechanics, and quantum field theory.  ...  Acknowledgements This work is taken from the author's Ph.D. thesis, done under the supervision of Persi Diaconis. His idea of studying the random partitions λ φ led to this work.  ... 
arXiv:math/9712236v1 fatcat:tnai4xvagbcyro6ht6jmwxafha

Page 3920 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews  
In a fundamental paper Alladi, Andrews and B. Gordon [J. Reine Angew.  ...  The proof of this “quartic key identity” (which does not follow from the earlier mentioned key identity) requires “only” the very-well-poised ¢s-summation.  ... 

A Probabilistic Proof of the Rogers Ramanujan Identities [article]

Jason Fulman
2000 arXiv   pre-print
Elementary probabilistic proofs of the Rogers-Ramanujan identities follow.  ...  The asymptotic probability theory of conjugacy classes of the finite general linear and unitary groups leads to a probability measure on the set of all partitions of natural numbers.  ...  The author thanks Persi Diaconis for discussions and a referee for correcting historical mistakes in the bibliography.  ... 
arXiv:math/0001078v4 fatcat:y7cotu2qurbvvntappbgdufa2y

An expectation formula with applications

Mingjin Wang
2011 Journal of Mathematical Analysis and Applications  
As applications of the expectation formula, we give a transformation formula and an expansion of Sears' 3 φ 2 transformation formula.  ...  Probability distribution W (x, q) Al-Salam and Verma q-integral Lebesgue's dominated convergence theorem Basic hypergeometric function 3 φ 2 In this paper, we derive an expectation formula of a random  ...  In particular, the author thanks the referee for help to improve the presentation of the paper.  ... 
doi:10.1016/j.jmaa.2011.01.044 fatcat:tkmm2eaxufc3fckyyki5aafomi

A q-multisum identity arising from finite chain ring probabilities [article]

Jehanne Dousse, Robert Osburn
2022 arXiv   pre-print
In this note, we prove a general identity between a q-multisum B_N(q) and a sum of N^2 products of quotients of theta functions.  ...  The q-multisum B_N(q) recently arose in the computation of a probability involving modules over finite chain rings.  ...  The first author was partially supported by ANR COMBINé ANR-19-CE48-0011 and the Impulsion grant of IdexLyon. The second author was partially supported by Enterprise Ireland cs20212030.  ... 
arXiv:2111.11123v2 fatcat:xlsys47ibvdzniadg3mawogvci

A $q$-Multisum Identity Arising from Finite Chain Ring Probabilities

Jehanne Dousse, Robert Osburn
2022 Electronic Journal of Combinatorics  
In this note, we prove a general identity between a $q$-multisum $B_N(q)$ and a sum of $N^2$ products of quotients of theta functions.  ...  The $q$-multisum $B_N(q)$ recently arose in the computation of a probability involving modules over finite chain rings.  ...  Acknowledgements The authors thank Ole Warnaar for pointing out references [11, 13, 14] , and the referee for a careful reading and helpful suggestions.  ... 
doi:10.37236/10691 fatcat:w55epca27fcvhcshyubm7g6v2m

Random matrix theory over finite fields: a survey [article]

Jason Fulman
2001 arXiv   pre-print
Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.  ...  First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups.  ...  The author received the financial support of an NSF Postdoctoral Fellowship.  ... 
arXiv:math/0003195v2 fatcat:szqnlbqcsvbtnknjnafpidalpu

Density of Free Modules over Finite Chain Rings [article]

Eimear Byrne, Anna-Lena Horlemann, Karan Khathuria, Violetta Weger
2022 arXiv   pre-print
In both cases, the density results can be bounded by the Andrews-Gordon identities. We also study the asymptotic behaviour of modules generated by random matrices over ℛ.  ...  In this paper we focus on modules over a finite chain ring ℛ of size q^s. We compute the density of free modules of ℛ^n, where we separately treat the asymptotics in n,q and s.  ...  Acknowledgments The authors want to thank Gianira N. Alfarano and the anonymous referee for fruitful discussions on the shape of a module.  ... 
arXiv:2106.09403v2 fatcat:z7mkxzgpmzbc5b2caenze6r5am

Random matrix theory over finite fields

Jason Fulman
2001 Bulletin of the American Mathematical Society  
Then we describe a probabilistic picture of conjugacy classes of the finite classical groups.  ...  The first part of this paper surveys generating functions methods in the study of random matrices over finite fields, explaining how they arose from theoretical need.  ...  The author received the financial support of an NSF Postdoctoral Fellowship and wrote this article at Stanford University.  ... 
doi:10.1090/s0273-0979-01-00920-x fatcat:dzka2lak3zh5xlky4cfhm47vby

Program Analysis of Probabilistic Programs [article]

Maria I. Gorinova
2022 arXiv   pre-print
While substantial work has been done both to formalise probabilistic programming and to improve efficiency of inference, there has been little work that makes use of the available program structure, by  ...  The techniques analyse a probabilistic program and adapt it to make inference more efficient, sometimes in a way that would have been tedious or impossible to do by hand.  ...  Gorinova, Andrew D. Gordon, and Charles Sutton Gorinova, Andrew D.  ... 
arXiv:2204.06868v1 fatcat:2dbonwruuzaopil4aijdeuz4mi

Schur's partition theorem and mixed mock modular forms [article]

Kathrin Bringmann, Karl Mahlburg
2013 arXiv   pre-print
answer to Andrews' Conjecture on the modularity of the Schur-type generating function.  ...  Furthermore, we also complete the second part of Andrews' speculation by determining the asymptotic behavior of these functions.  ...  The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp Foundation and an individual research grant from the ERC.  ... 
arXiv:1307.1800v1 fatcat:qv3gbtqlufgebogcmlp5ac3c4a
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