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Asymptotic analogs of the Rogers-Ramanujan identities

Charles H Brenner
1986 Journal of combinatorial theory. Series A  
Andrew [2] has suggested using asymptotics to narrow the search for identities of the Rogers-Ramanujan type.  ...  +x15n+8) as x-+l, n=l (n, 15) = I which can be proved using the results of Section 1. Such formulas may be thought of as examples of Rogers-Ramanujan type asymptotic identities.  ... 
doi:10.1016/0097-3165(86)90069-5 fatcat:riertghb6barxjzg2njiysn7ii

Bounds for d-distinct partitions

Soon-Yi Kang, Young Kim
2021 Hardy-Ramanujan Journal  
International audience Euler's identity and the Rogers-Ramanujan identities are perhaps the most famous results in the theory of partitions.  ...  In this note, we present the recent developments of generalizations and analogs of the Alder-Andrews Theorem and establish asymptotic lower and upper bounds for the d-distinct partitions.  ...  The authors are very grateful to the referee for careful reading of the paper and helpful comments.  ... 
doi:10.46298/hrj.2021.7430 fatcat:le576algt5fgnkiho5y652eftm

Book Review: Ramanujan's lost notebook, Part I

David M. Bressoud
2006 Bulletin of the American Mathematical Society  
Their asymptotic series for the number of partitions of an integer, published in 1918 [32], later refined by Rademacher [35] into a rapidly convergent series, remains one of the great achievements of analytic  ...  Hardy credited Ramanujan for all of the inspiration. Nevertheless, even Hardy expressed uncertainty about the true greatness of Ramanujan's accomplishments:  ...  Chapters 9-11 deal with q-series: transformations that follow from the Rogers-Fine identity, empirical evidence for the Rogers-Ramanujan identities including comparison of asymptotics, and extensions to  ... 
doi:10.1090/s0273-0979-06-01110-4 fatcat:ma346xa4yzhozdh6zfog25wqvq

Page 3043 of Mathematical Reviews Vol. , Issue 84h [page]

1984 Mathematical Reviews  
An easy proof of the Rogers- Ramanujan identities. J. Number Theory 16 (1983), no. 2, 235-241. The author gives a simple analytical proof of the Rogers- Ramanujan identities [G. E.  ...  to the first Rogers-Ramanujan identity in the standard way.  ... 

An Infinite Family of Engel Expansions of Rogers–Ramanujan Type

George E. Andrews, Arnold Knopfmacher, Peter Paule
2000 Advances in Applied Mathematics  
Various examples related to classical partition theorems, including the Rogers-Ramanujan identities, have been given recently.  ...  The object of this paper is to show that the new and elegant Rogers-Ramanujan generalization found by Garrett, Ismail, and Stanton also fits into this framework.  ...  We conclude by recalling the classic Rogers-Ramanujan identities, namely the instances m = 0 and m = 1 of (8).  ... 
doi:10.1006/aama.2000.0686 fatcat:rljwopimjrfbbc5grbacqcq5oi

Page 295 of Mathematical Reviews Vol. 20, Issue 3 [page]

1959 Mathematical Reviews  
The proofs of the two results are also analogous. F. V. Atkinson (Canberra) where A= 1803: Singh, V. N. Certain generalized hypergeometric iden- tities of the Rogers-Ramanujan type. [I. Pacific J.  ...  (cos 6’)P—»,s4(cos 0’) over —oo< s<oo, the result being analogous to a Bessel-function formula of Ramanujan [see G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed.  ... 

Page 4348 of Mathematical Reviews Vol. , Issue 94h [page]

1994 Mathematical Reviews  
Using the analogy with bound- ary value problems for partial differential equations, we can obtain identities for Fo. In particular, we derive Rogers-Ramanujan in a ‘natural’ fasion again.  ...  Section V explains one of the pitfalls one encounters in attempt- ing to prove the Rogers-Ramanujan identities by comparing the 11 NUMBER THEORY singularities of the generating functions.”  ... 

Page 1203 of Mathematical Reviews Vol. , Issue 96b [page]

1996 Mathematical Reviews  
In this case, the authors find an A, x E> structure and prove a Rogers-Ramanujan identity of A, x E7 type.  ...  In the thermodynamic limit, these polynomial identities yield a proof of the Eg Rogers-Ramanujan identity recently conjectured by R. Kedem et al. [Phys. Lett.  ... 

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

Page 3560 of Mathematical Reviews Vol. , Issue 2004e [page]

2004 Mathematical Reviews  
By using three Rogers-Ramanujan type identities from Slater’s list [L. J. Slater, Proc. London Math.  ...  Soc. (2) 54 (1952), 147-167; MR 14, 138e], the authors deduce three Rogers-Ramanujan type partition identities for Burge’s restricted partition pairs [W. H. Burge, J. Combin. Theory Ser.  ... 

Rogers–Ramanujan computer searches

James McLaughlin, Andrew V. Sills, Peter Zimmer
2009 Journal of symbolic computation  
We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers-Ramanujan type and identities of false theta functions  ...  Richmond and Szekeres (1981) applied asymptotic analysis to two Rogers-Ramanujan type identities to derive identities for the function L(x).  ...  identity deriving from an identity of Rogers-Ramanujan type.  ... 
doi:10.1016/j.jsc.2009.02.003 fatcat:yqm6gshcf5cnlap4iimehx3jpy

On partition functions related to Schur's second partition theorem

George E. Andrews
1968 Proceedings of the American Mathematical Society  
The first Rogers-Ramanujan identity [5, p. 291 ] asserts that Bi(n) is equal to the number of partitions of n into parts = +1 (mod 5).  ...  The second Rogers-Ramanujan identity asserts that Ci(n) is equal to the number of partitions of n into parts = ±2 (mod 5). A theorem proved independently by H. Gollnitz [3] and B.  ... 
doi:10.1090/s0002-9939-1968-0225741-2 fatcat:27tfzstrdbcj5bnboxpzasyoe4


1996 International Journal of Modern Physics A  
In support of our conjectures we establish the correct behaviour under level-rank duality for G=A_n-1 and show that the A-D-E Rogers--Ramanujan identities have the expected q→ 1^- asymptotics in terms  ...  We conjecture polynomial identities which imply Rogers--Ramanujan type identities for branching functions associated with the cosets ( G^(1))_ℓ-1⊗ ( G^(1))_1 / ( G^(1))_ℓ, with G=A_n-1 (ℓ≥ 2), D_n-1 (ℓ  ...  Acknowledgements We a c knowledge many helpful and stimulating discussions on Rogers{Ramanujan identities with Omar Foda.  ... 
doi:10.1142/s0217751x96000146 fatcat:n2bqeahedbavvfrpyg3a4z2vse

Page 138 of Mathematical Reviews Vol. 14, Issue 2 [page]

1953 Mathematical Reviews  
Further identities of the Rogers-Ramanujan type. Proc. London Math. Soc. (2) 54, 147-167 (1952).  ...  The author has recently outlined a method for obtaining identities of the Rogers-Ramanujan type [same Proc. 53, 460-475 (1951); these Rev. 13, 227].  ... 

Page 7661 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
The first author conjectured explicit formulas for P,,(q) and T,,(q) in [“Computer algebra and identities of the Rogers-Ramanujan type”, Ph.D. the- sis, Pennsylvania State Univ., 1991], and the authors  ...  The authors begin with the following pair of identities, due to L. J. Rogers, which appear as identities 94 and 99, respectively, in [L. J. Slater, Proc. London Math.  ... 
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