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Asymptotic analogs of the Rogers-Ramanujan identities

1986
*
Journal of combinatorial theory. Series A
*

Andrew [2] has suggested using

doi:10.1016/0097-3165(86)90069-5
fatcat:riertghb6barxjzg2njiysn7ii
*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*. ...##
###
Bounds for d-distinct partitions

2021
*
Hardy-Ramanujan Journal
*

International audience Euler's

doi:10.46298/hrj.2021.7430
fatcat:le576algt5fgnkiho5y652eftm
*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. ...##
###
Book Review: Ramanujan's lost notebook, Part I

2006
*
Bulletin of the American Mathematical Society
*

Their

doi:10.1090/s0273-0979-06-01110-4
fatcat:ma346xa4yzhozdh6zfog25wqvq
*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 ...##
###
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

2000
*
Advances in Applied Mathematics
*

Various examples related to classical partition theorems, including

doi:10.1006/aama.2000.0686
fatcat:rljwopimjrfbbc5grbacqcq5oi
*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). ...##
###
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]

2000
*
arXiv
*
pre-print

Elementary probabilistic proofs

arXiv:math/0001078v4
fatcat:y7cotu2qurbvvntappbgdufa2y
*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. ...##
###
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

2009
*
Journal of symbolic computation
*

We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to

doi:10.1016/j.jsc.2009.02.003
fatcat:yqm6gshcf5cnlap4iimehx3jpy
*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. ...##
###
On partition functions related to Schur's second partition theorem

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. ...

##
###
A-D-E POLYNOMIAL AND ROGERS-RAMANUJAN IDENTITIES

1996
*
International Journal of Modern Physics A
*

In support

doi:10.1142/s0217751x96000146
fatcat:n2bqeahedbavvfrpyg3a4z2vse
*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. ...##
###
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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