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Journal articles on the topic 'Identités de Rogers-Ramanujan'

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Published: 4 June 2021

Last updated: 5 February 2022

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1

SILLS, ANDREW V. "IDENTITIES OF THE ROGERS–RAMANUJAN–SLATER TYPE." International Journal of Number Theory 03, no. 02 (2007): 293–323. http://dx.doi.org/10.1142/s1793042107000912.

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It is shown that (two-variable generalizations of) more than half of Slater's list of 130 Rogers–Ramanujan identities (L. J. Slater, Further identities of the Rogers–Ramanujan type, Proc. London Math Soc. (2)54 (1952) 147–167) can be easily derived using just three multiparameter Bailey pairs and their associated q-difference equations. As a bonus, new Rogers–Ramanujan type identities are found along with natural combinatorial interpretations for many of these identities.

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2

Hossain, Fazlee, Sabuj Das, and Haradhan Kumar Mohajan. "The Rogers-Ramanujan Identities." Turkish Journal of Analysis and Number Theory 3, no. 2 (2016): 37–42. http://dx.doi.org/10.12691/tjant-3-2-1.

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3

Guo, Victor J. W., Frédéric Jouhet, and Jiang Zeng. "New finite Rogers-Ramanujan identities." Ramanujan Journal 19, no. 3 (2009): 247–66. http://dx.doi.org/10.1007/s11139-009-9182-0.

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4

WARNAAR, S. OLE, and PAUL A. PEARCE. "A-D-E POLYNOMIAL AND ROGERS-RAMANUJAN IDENTITIES." International Journal of Modern Physics A 11, no. 02 (1996): 291–311. http://dx.doi.org/10.1142/s0217751x96000146.

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We conjecture polynomial identities which imply Rogers-Ramanujan type identities for branching functions associated with the cosets [Formula: see text], with [Formula: see text], Dn–1 (ℓ≥2), E6,7,8 (ℓ=2). In support of our conjectures we establish the correct behavior under level-rank duality for [Formula: see text] and show that the A–D–E Rogers-Ramanujan identities have the expected q→1− asymptotics in terms of dilogarithm identities. Possible generalizations to arbitrary cosets are also discussed briefly.

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5

Agarwal, A. K., and M. Goyal. "New Partition Theoretic Interpretations of Rogers-Ramanujan Identities." International Journal of Combinatorics 2012 (May 13, 2012): 1–6. http://dx.doi.org/10.1155/2012/409505.

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The generating function for a restricted partition function is derived. This in conjunction with two identities of Rogers provides new partition theoretic interpretations of Rogers-Ramanujan identities.

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6

JING, NAIHUAN, KAILASH C. MISRA, and CARLA D. SAVAGE. "ON MULTI-COLOR PARTITIONS AND THE GENERALIZED ROGERS–RAMANUJAN IDENTITIES." Communications in Contemporary Mathematics 03, no. 04 (2001): 533–48. http://dx.doi.org/10.1142/s0219199701000482.

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Basil Gordon, in the sixties, and George Andrews, in the seventies, generalized the Rogers–Ramanujan identities to higher moduli. These identities arise in many areas of mathematics and mathematical physics. One of these areas is representation theory of infinite dimensional Lie algebras, where various known interpretations of these identities have led to interesting applications. Motivated by their connections with Lie algebra representation theory, we give a new interpretation of a sum related to generalized Rogers–Ramanujan identities in terms of multi-color partitions.

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7

MEURMAN, ARNE, and MIRKO PRIMC. "A BASIS OF THE BASIC $\mathfrak{sl} ({\bf 3}, {\mathbb C})^~$-MODULE." Communications in Contemporary Mathematics 03, no. 04 (2001): 593–614. http://dx.doi.org/10.1142/s0219199701000512.

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J. Lepowsky and R. L. Wilson initiated the approach to combinatorial Rogers–Ramanujan type identities via the vertex operator constructions of representations of affine Lie algebras. In this approach the first new combinatorial identities were discovered by S. Capparelli through the construction of the level 3 standard [Formula: see text]-modules. We obtained several infinite series of new combinatorial identities through the construction of all standard [Formula: see text]-modules; the identities associated to the fundamental modules coincide with the two Capparelli identities. In this paper we extend our construction to the basic [Formula: see text]-module and, by using the principal specialization of the Weyl–Kac character formula, we obtain a Rogers–Ramanujan type combinatorial identity for colored partitions. The new combinatorial identity indicates the next level of complexity which one should expect in Lepowsky–Wilson's approach for affine Lie algebras of higher ranks, say for [Formula: see text], n ≥ 2, in a way parallel to the next level of complexity seen when passing from the Rogers–Ramanujan identities (for modulus 5) to the Gordon identities for odd moduli ≥7.

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8

Adiga, Chandrashekar, A. Vanitha, and M. S. Surekha. "Continued Fractions of Order Six and New Eisenstein Series Identities." Journal of Numbers 2014 (May 27, 2014): 1–6. http://dx.doi.org/10.1155/2014/643241.

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We prove two identities for Ramanujan’s cubic continued fraction and a continued fraction of Ramanujan, which are analogues of Ramanujan’s identities for the Rogers-Ramanujan continued fraction. We further derive Eisenstein series identities associated with Ramanujan’s cubic continued fraction and Ramanujan’s continued fraction of order six.

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9

O’Hara, Kathleen, and Dennis Stanton. "Refinements of the Rogers–Ramanujan Identities." Experimental Mathematics 24, no. 4 (2015): 410–18. http://dx.doi.org/10.1080/10586458.2015.1014076.

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10

Garrett, Kristina, Mourad E. H. Ismail, and Dennis Stanton. "Variants of the Rogers–Ramanujan Identities." Advances in Applied Mathematics 23, no. 3 (1999): 274–99. http://dx.doi.org/10.1006/aama.1999.0658.

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11

Biagioli, Anthony J. F. "A Proof of some identities of Ramanujan using modular forms." Glasgow Mathematical Journal 31, no. 3 (1989): 271–95. http://dx.doi.org/10.1017/s0017089500007850.

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In 1974 B. J. Birch [1] published a description of some manuscripts of Ramanujan which contained, among other things, a list of forty identities involving the Rogers-Ramanujan functionsAt that time nine of these had been proven, and since then twenty-two more of them have been proven, fifteen of them by David Bressoud in his thesis [2]. Bressoud gives a synopsis of the extant proofs, where he attributes proofs to H. B. C. Darling [3], L. J. Rogers [4], L. J. Mordell [5], and G. N. Watson [6].

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12

Chan, Hei-Chi. "Revisiting a Classic Identity That Implies the Rogers–Ramanujan Identities II." Axioms 10, no. 4 (2021): 239. http://dx.doi.org/10.3390/axioms10040239.

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We give a new proof of an identity due to Ramanujan. From this identity, he deduced the famous Rogers–Ramanujan identities. We prove this identity by establishing a simple recursion Jk=qkJk−1, where |q|<1. This is a sequel to our recent work.

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13

FODA, OMAR, and YAS-HIRO QUANO. "POLYNOMIAL IDENTITIES OF THE ROGERS-RAMANUJAN TYPE." International Journal of Modern Physics A 10, no. 16 (1995): 2291–315. http://dx.doi.org/10.1142/s0217751x9500111x.

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Presented are polynomial identities which imply generalizations of Euler and Rogers-Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by establishing a graphical one-to-one correspondence between those two kinds of restricted partitions.

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14

TERHOEVEN, MICHAEL. "DILOGARITHM IDENTITIES, FUSION RULES AND STRUCTURE CONSTANTS OF CFTs." Modern Physics Letters A 09, no. 02 (1994): 133–41. http://dx.doi.org/10.1142/s0217732394000149.

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Recently dilogarithm identities have made their appearance in the physics literature. These identities seem to allow to calculate structure constants like, in particular, the effective central charge of certain conformal field theories from their fusion rules. In Ref. 12 a proof of identities of this type was given by considering the asymptotics of character functions in the so-called Rogers-Ramanujan sum form and comparing with the asymptotics predicted by modular covariance. Refining the argument, we obtain the general connection of quantum dimensions of certain conformal field theories to the arguments of the dilogarithm function in the identities in question and an infinite set of consistency conditions on the parameters of Rogers-Ramanujan type partitions for them to be modular covariant.

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15

Baruah, Nayandeep Deka, and P. Bhattacharyya. "Some theorems on the explicit evaluation of Ramanujan's theta-functions." International Journal of Mathematics and Mathematical Sciences 2004, no. 40 (2004): 2149–59. http://dx.doi.org/10.1155/s0161171204111058.

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Bruce C. Berndt et al. and Soon-Yi Kang have proved many of Ramanujan's formulas for the explicit evaluation of the Rogers-Ramanujan continued fraction and theta-functions in terms of Weber-Ramanujan class invariants. In this note, we give alternative proofs of some of these identities of theta-functions recorded by Ramanujan in his notebooks and deduce some formulas for the explicit evaluation of his theta-functions in terms of Weber-Ramanujan class invariants.

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16

Kožić, Slaven, та Mirko Primc. "Quasi-particles in the principal picture of 𝔰𝔩̂2 and Rogers–Ramanujan-type identities". Communications in Contemporary Mathematics 20, № 05 (2018): 1750073. http://dx.doi.org/10.1142/s0219199717500730.

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In their seminal work Lepowsky and Wilson gave a vertex-operator theoretic interpretation of Gordon–Andrews–Bressoud’s generalization of Rogers–Ramanujan combinatorial identities, by constructing bases of vacuum spaces for the principal Heisenberg subalgebra of standard [Formula: see text]-modules, parametrized with partitions satisfying certain difference 2 conditions. In this paper, we define quasi-particles in the principal picture of [Formula: see text] and construct quasi-particle monomial bases of standard [Formula: see text]-modules for which principally specialized characters are given as products of sum sides of the corresponding analytic Rogers–Ramanujan-type identities with the character of the Fock space for the principal Heisenberg subalgebra.

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17

Genish, Arel, and Doron Gepner. "Generalized Rogers–Ramanujan identities for twisted affine algebras." Modern Physics Letters A 32, no. 21 (2017): 1750110. http://dx.doi.org/10.1142/s0217732317501103.

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The characters of parafermionic conformal field theories are given by the string functions of affine algebras, which are either twisted or untwisted algebras. Expressions for these characters as generalized Rogers–Ramanujan algebras have been established for the untwisted affine algebras. However, we study the identities for the string functions of the twisted affine Lie algebras. A conjecture for the string functions was proposed by Hatayama et al., for the unit fields, which expresses the string functions as Rogers–Ramanujan type sums. Here we propose to check the Hatayama et al. conjecture, using Lie algebraic theoretic methods. We use Freudenthal’s formula, which we computerized, to verify the identities for all the algebras at low rank and low level. We find complete agreement with the conjecture.

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18

Berndt, Bruce C., and Hamza Yesilyurt. "New identities for the Rogers–Ramanujan functions." Acta Arithmetica 120, no. 4 (2005): 395–413. http://dx.doi.org/10.4064/aa120-4-5.

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19

Chu, Wenchang, and Wenlong Zhang. "Iteration method for multiple Rogers-Ramanujan identities." Kodai Mathematical Journal 32, no. 3 (2009): 471–500. http://dx.doi.org/10.2996/kmj/1257948891.

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20

Keilthy, Adam, and Robert Osburn. "Rogers–Ramanujan type identities for alternating knots." Journal of Number Theory 161 (April 2016): 255–80. http://dx.doi.org/10.1016/j.jnt.2015.02.002.

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21

Warnaar, S. Ole, та Wadim Zudilin. "Dedekind's η-function and Rogers-Ramanujan identities". Bulletin of the London Mathematical Society 44, № 1 (2012): 1–11. http://dx.doi.org/10.1112/blms/bdr019.

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22

Andrews, George E., Arnold Knopfmacher, and John Knopfmacher. "Engel Expansions and the Rogers–Ramanujan Identities." Journal of Number Theory 80, no. 2 (2000): 273–90. http://dx.doi.org/10.1006/jnth.1999.2453.

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23

Alladi, Krishnaswami. "An Invitation to the Rogers-Ramanujan Identities." Notices of the American Mathematical Society 67, no. 01 (2020): 1. http://dx.doi.org/10.1090/noti2013.

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24

Sills, Andrew V. "On identities of the Rogers-Ramanujan type." Ramanujan Journal 11, no. 3 (2006): 403–29. http://dx.doi.org/10.1007/s11139-006-8483-9.

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25

Bruschek, Clemens, Hussein Mourtada, and Jan Schepers. "Arc spaces and the Rogers–Ramanujan identities." Ramanujan Journal 30, no. 1 (2012): 9–38. http://dx.doi.org/10.1007/s11139-012-9401-y.

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26

Chu, W., and C. Wang. "Iteration process for multiple rogers–ramanujan identities." Ukrainian Mathematical Journal 64, no. 1 (2012): 110–39. http://dx.doi.org/10.1007/s11253-012-0633-1.

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27

Nambiar, K. K. "A graphic illustration of Rogers-Ramanujan Identities." Applied Mathematics Letters 9, no. 5 (1996): 11–15. http://dx.doi.org/10.1016/0893-9659(96)00065-1.

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28

Gepner, Doron. "Lattice models and generalized Rogers-Ramanujan identities." Physics Letters B 348, no. 3-4 (1995): 377–85. http://dx.doi.org/10.1016/0370-2693(95)00173-i.

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29

Bhatnagar, Gaurav. "How to discover the Rogers-Ramanujan identities." Resonance 20, no. 5 (2015): 416–30. http://dx.doi.org/10.1007/s12045-015-0199-y.

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30

Agarwal, A. K. "Rogers-Ramanujan identities for n-color partitions." Journal of Number Theory 28, no. 3 (1988): 299–305. http://dx.doi.org/10.1016/0022-314x(88)90045-5.

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31

Brenner, Charles H. "Asymptotic analogs of the Rogers-Ramanujan identities." Journal of Combinatorial Theory, Series A 43, no. 2 (1986): 303–19. http://dx.doi.org/10.1016/0097-3165(86)90069-5.

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32

Paule, Peter. "On identities of the Rogers-Ramanujan type." Journal of Mathematical Analysis and Applications 107, no. 1 (1985): 255–84. http://dx.doi.org/10.1016/0022-247x(85)90368-3.

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33

Chu, Wenchang, and Wenlong Zhang. "Bilateral Bailey lemma and Rogers–Ramanujan identities." Advances in Applied Mathematics 42, no. 3 (2009): 358–91. http://dx.doi.org/10.1016/j.aam.2008.07.003.

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34

Sills, Andrew V. "Identities of the Rogers–Ramanujan–Bailey type." Journal of Mathematical Analysis and Applications 308, no. 2 (2005): 669–88. http://dx.doi.org/10.1016/j.jmaa.2004.11.061.

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35

Corteel, Sylvie, and Olivier Mallet. "Overpartitions, lattice paths, and Rogers–Ramanujan identities." Journal of Combinatorial Theory, Series A 114, no. 8 (2007): 1407–37. http://dx.doi.org/10.1016/j.jcta.2007.02.004.

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36

Cherednik, Ivan, and Boris Feigin. "Rogers–Ramanujan type identities and Nil-DAHA." Advances in Mathematics 248 (November 2013): 1050–88. http://dx.doi.org/10.1016/j.aim.2013.08.025.

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37

Wang, Chun, and Shane Chern. "Some q-transformation formulas and Hecke type identities." International Journal of Number Theory 15, no. 07 (2019): 1349–67. http://dx.doi.org/10.1142/s1793042119500751.

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In this paper, we establish certain transformations on basic hypergeometric series. Some applications of these transformation formulas to Hecke type identities will be discussed. We also study other [Formula: see text]-series transformations that may lead to certain Rogers–Ramanujan type identities.

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38

Schilling, Anne, and S. Ole Warnaar. "A Higher-Level Bailey Lemma." International Journal of Modern Physics B 11, no. 01n02 (1997): 189–95. http://dx.doi.org/10.1142/s0217979297000253.

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We propose a generalization of Bailey's lemma, useful for proving q-series identities. As an application, generalizations of Euler's identity, the Rogers–Ramanujan identities, and the Andrews–Gordon identities are derived. This generalized Bailey lemma also allows one to derive the branching functions of higher-level [Formula: see text] cosets.

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39

ANDREWS, GEORGE E. "A FINE DREAM." International Journal of Number Theory 03, no. 03 (2007): 325–34. http://dx.doi.org/10.1142/s1793042107000948.

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We shall develop further N. J. Fine's theory of three parameter non-homogeneous first order q-difference equations. The object of our work is to bring the Rogers–Ramanujan identities within the purview of such a theory. In addition, we provide a number of new identities.

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40

BOWMAN, D., J. Mc LAUGHLIN, and N. J. WYSHINSKI. "A q-CONTINUED FRACTION." International Journal of Number Theory 02, no. 04 (2006): 523–47. http://dx.doi.org/10.1142/s179304210600070x.

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We use the method of generating functions to find the limit of a q-continued fraction, with 4 parameters, as a ratio of certain q-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for (q2; q3)∞/(q; q3)∞and [Formula: see text]. In addition, we give a new proof of the famous Rogers–Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.

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41

Bernd, Bruce C., and Heng Huat Chan. "Some Values for the Rogers-Ramanujan Continued Fraction." Canadian Journal of Mathematics 47, no. 5 (1995): 897–914. http://dx.doi.org/10.4153/cjm-1995-046-5.

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AbstractIn his first and lost notebooks, Ramanujan recorded several values for the Rogers-Ramanujan continued fraction. Some of these results have been proved by K. G. Ramanathan, using mostly ideas with which Ramanujan was unfamiliar. In this paper, eight of Ramanujan's values are established; four are proved for the first time, while the remaining four had been previously proved by Ramanathan by entirely different methods. Our proofs employ some of Ramanujan's beautiful eta-function identities, which have not been heretofore used for evaluating continued fractions.

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42

FULMAN, JASON. "A PROBABILISTIC PROOF OF THE ROGERS–RAMANUJAN IDENTITIES." Bulletin of the London Mathematical Society 33, no. 4 (2001): 397–407. http://dx.doi.org/10.1017/s0024609301008207.

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The asymptotic probability theory of conjugacy classes of the finite general groups leads to a probability measure on the set of all partitions of natural numbers. A simple method of understanding these measures in terms of Markov chains is given in this paper, leading to an elementary probabilistic proof of the Rogers–Ramanujan identities. This is compared with work on the uniform measure. The main case of Bailey's lemma is interpreted as finding eigenvectors of the transition matrix of a Markov chain. It is shown that the viewpoint of Markov chains extends to quivers.

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43

Prodinger, Helmut. "Pseudoq-Engel expansions and Rogers-Ramanujan type identities." Quaestiones Mathematicae 35, no. 1 (2012): 23–33. http://dx.doi.org/10.2989/16073606.2012.671164.

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44

Andrews, George E., and R. J. Baxter. "A Motivated Proof of the Rogers-Ramanujan Identities." American Mathematical Monthly 96, no. 5 (1989): 401. http://dx.doi.org/10.2307/2325145.

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45

Andrews, George E., and R. J. Baxter. "A Motivated Proof of the Rogers-Ramanujan Identities." American Mathematical Monthly 96, no. 5 (1989): 401–9. http://dx.doi.org/10.1080/00029890.1989.11972207.

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46

Bos, M. K., and K. C. Misra. "Level two representations of and Rogers-Ramanujan identities." Communications in Algebra 22, no. 10 (1994): 3965–83. http://dx.doi.org/10.1080/00927879408825059.

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47

Ismail, M. E. H., and D. Stanton. "Tribasic integrals and identities of Rogers-Ramanujan type." Transactions of the American Mathematical Society 355, no. 10 (2003): 4061–91. http://dx.doi.org/10.1090/s0002-9947-03-03338-5.

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48

Berndt, Bruce C., Geumlan Choi, Youn-Seo Choi, et al. "Ramanujan’s forty identities for the Rogers-Ramanujan functions." Memoirs of the American Mathematical Society 188, no. 880 (2007): 0. http://dx.doi.org/10.1090/memo/0880.

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49

Ole Warnaar, S. "Partial-Sum Analogues of the Rogers–Ramanujan Identities." Journal of Combinatorial Theory, Series A 99, no. 1 (2002): 143–61. http://dx.doi.org/10.1006/jcta.2002.3266.

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50

Bowman, Douglas, James Mc Laughlin, and Andrew V. Sills. "Some more identities of the Rogers-Ramanujan type." Ramanujan Journal 18, no. 3 (2008): 307–25. http://dx.doi.org/10.1007/s11139-007-9109-6.

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