Relevant bibliographies by topics / Identités de Rogers-Ramanujan

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Academic literature on the topic 'Identités de Rogers-Ramanujan'

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Published: 4 June 2021
Last updated: 5 February 2022

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

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Identités de Rogers-Ramanujan"

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Mallet, Olivier. "Autour des surpartitions et des identités de type Rogers-Ramanujan." Phd thesis, Université Paris-Diderot - Paris VII, 2008. http://tel.archives-ouvertes.fr/tel-00366067.

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Une partition d'un entier positif est une façon d'écrire ce nombre comme une somme d'entiers strictement positifs où l'ordre des termes ne compte pas. Plusieurs généralisations des partitions ont été étudiées, parmi lesquelles les surpartitions, qui sont des partitions où l'on peut surligner la dernière occurrence d'un nombre, les paires de surpartitions ou encore les partitions n-colorées, qui sont liées à un modèle de physique statistique. Dans cette thèse, on généralise aux paires de surpartitions les identités d'Andrews-Gordon, qui sont une extension d'un résultat classique de la théorie des partitions : les identités de Rogers-Ramanujan. Pour cela, on définit deux classes de séries hypergéométriques basiques et on montre que ce sont les séries génératrices des paires de surpartitions vérifiant différents types de conditions (multiplicités, rangs successifs, dissection de Durfee) et de certains chemins du plan. On montre également que pour certaines valeurs des paramètres, ces séries peuvent s'écrire comme des produits infinis, ce qui conduit à plusieurs identités de type Rogers-Ramanujan. La démonstration utilise diverses méthodes combinatoires et analytiques. On définit enfin une généralisation des partitions n-colorées, les surpartitions n-colorées, et on les utilise pour interpréter combinatoirement certaines séries multiples et démontrer d'autres identités de type Rogers-Ramanujan.
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Konan, Isaac. "Rogers-Ramanujan type identities : bijective proofs and Lie-theoretic approach." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7087.

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Cette thèse relève de la théorie des partitions d’entiers, à l’intersection de la combinatoire et de la théorie de nombres. En particulier, nous étudions les identités de type Rogers-Ramanujan sous le spectre de la méthode des mots pondérés. Une révision de cette méthode nous permet d’introduire de nouveaux objets combinatoires au delà de la notion classique de partitions d’entiers: partitions colorées généralisées. À l’aide de ces nouveaux éléments, nous établissons de nouvelles identités de type Rogers-Ramanujanvia deux approches différentes. La première approche consiste en une preuve combinatoire, essentiellement bijective, des identités étudiées. Cette approche nous a ainsi permis d’établir des identités généralisant plusieurs identités importantes de la théorie: l’identité de Schur et l’identité Göllnitz, l’identité de Glaisher généralisant l’identité d’Euler, les identités de Siladić, de Primc et de Capparelli issues de la théorie des représentations de algèbres de Lie affines. La deuxième approche fait appel à la théorie des cristaux parfaits, issue de la théorie des représentations des algèbres de Lie affines. Nous interprétons ainsi le caractère des représentations standards comme des identités de partitions d’entiers colorées généralisées. En particulier, cette approche permet d’établir des formules assez simplifiées du caractère pour toutes les représentations standards de niveau 1 des types affines A(1) n-1, A(2) 2n , D(2) n+1, A(2) 2n-1, B(1) n , D(1) n<br>The topic of this thesis belongs to the theory of integer partitions, at the intersection of combinatorics and number theory. In particular, we study Rogers-Ramanujan type identities in the framework of the method of weighted words. This method revisited allows us to introduce new combinatorial objects beyond the classical notion of integer partitions: the generalized colored partitions. Using these combinatorial objects, we establish new Rogers-Ramanujan identities via two different approaches.The first approach consists of a combinatorial proof, essentially bijective, of the studied identities. This approach allowed us to establish some identities generalizing many important identities of the theory of integer partitions : Schur’s identity and Göllnitz’ identity, Glaisher’s identity generalizing Euler’s identity, the identities of Siladić, of Primc and of Capparelli coming from the representation theory of affine Lie algebras. The second approach uses the theory of perfect crystals, coming from the representation theory of affine Lie algebras. We view the characters of standard representations as some identities on the generalized colored partitions. In particular, this approach allows us to establish simple formulas for the characters of all the level one standard representations of type A(1) n-1, A(2) 2n , D(2) n+1, A(2) 2n-1, B(1) n , D(1) n
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Afsharijoo, Pooneh. "Looking for a new version of Gordon's identities : from algebraic geometry to combinatorics through partitions." Thesis, Sorbonne Paris Cité, 2019. https://theses.md.univ-paris-diderot.fr/AFSHARIJOO_Pooneh_2_complete_20190510.pdf.

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Une partition d’un nombre entier positif n est une suite décroissante des entiers positifs dont la somme est égal à n. Les entiers qui y apparaissent sont appelés les parties de la partition. Ma thèse est centrée sur l’étude des partitions des nombres entiers et les identités qui les relient. Plus précisément, il s’agit de montrer que le nombre de partitions ayant une propriété A est égal au nombre de partitions ayant une autre propriété B. Ce type d’identité joue un rôle important en théorie des nombres, en combinatoire, en théorie de représentations et en physique statistique. Une de ces identités est la suivante: Théorème. (La première identité de Rogers-Ramanujan) Le nombre de partitions d’un nombre naturel n dont les parties sont congruentes à 1 ou 4 modulo 5 est égal au nombre de partitions de n dont les parties ne sont ni égales ni consécutives. Dans ce travail, on étudie les identités entre les partitions en utilisant la relation entre les combinatoires des partitions et les combinatoires de l’algèbre graduée associée à un objet important de la géométrie algèbrique: l’espace des arcs. Étant donnés un corps k de caractéristique zéro et des polynômes f1,…,fm de k[x1,…, xn], l’espace des arcs associé correspond à l’idéal I de S:=k[x1j,…, xnj|j&gt;-1], engendré par les coefficients de certains développements associés aux polynômes ci-dessus et aux variables xij. Si on prend xi,0 = 0 pour i=1,…,n, la série de Hilbert-Poincaré de l’algèbre graduée S\I est étroitement liée aux partitions des entiers satisfaisants des conditions qui dépendent de l’idéal I. Dans le cas où f(x) = x^r de k[x], l’idéal I de k[x1, x2, … ] est un idéal différentiel pour la dérivation D(xi) = xi+1, dans le sens que DI est inclus dans I. En effet, dans ce cas I est engendré par x1^r et tous ses dérivés itérées. nous montrons que pour r = 2 le calcul de la base de Gröbner de l’ideal I par rapport à l’ordre lexicographique pondéré est lié à une identité faisant intervenir les partitions qui apparaissent dans la première identité de Rogers-Ramanujan. Nous prouvons ensuite qu’une base de Gröbner de cet idéal n’est pas différentiellement finie, au contraire du cas de l’ordre lexicographique inverse pondéré. Nous donnons une preuve de ce point de vue des identités de Gordon qui forment une famille importante d’identités reliant les partitions. En utilisant des idéaux différentiels et des méthodes venant de l’espace des arcs, nous énonçons une conjecture qui pourrait ajouter un nouveau membre aux identités de Gordon. Nous l’avons déjà démontré pour un cas particulier. À la fin, nous donnons une preuve simple et directe d’un théorème de Nguyen Duc Tam sur la base de Gröbner de l’idéal différentiel [x1y1]; Nous obtenons ensuite des identités entres les partitions avec 2 couleurs<br>A partition of a positive integer n is a decreasing sequence of positive integers such that their sum is equal to n. The integers which appear in this sequence are called the parts of this partition. My thesis studies the partitions of integers and the identities between them. A partition identity is an equality between the number of partitions of an integer n satisfying a certain condition A and the number of partitions of n satisfying another condition B. They play an important role in many areas: number theory, combinatorics, Lie theory, particle physics and statistical mechanics. One of these identities is as follows: Theorem. (The first Rogers-Ramanujan identity) The number of partitions of a positive integer n with no equal or consecutive parts is equal to the number of partitions of n into parts 1 or 4(mod.5). In this work, we study partition identities using the relation between the combinatorics of partitions and the combinatorics of graded algebras associated to an important object of algebraic geometry: arc spaces. Given a field k of characteristic zero and polynomials f1,…,fm in k[x1,…, xn], the associated arc space is the space corresponds to the ideal I of S:=k[x1j,…, xnj|j&gt;-1], generated by the coefficients of some developments associated to the above polynomials and the variables xij. For focussed arcs, which is when we take xi0 = 0 for i=1,…,n, the Hilbert-Poincaré series of the graded algebra S\I is closely related to partitions of integers satisfying conditions depending on I. In the case where f(x) = x^r in k[x], the ideal I of k[x1, x2,… ] is a differential ideal for the derivation D(xi) = xi+1, in the sens that DI is included in I. In fact it is generated by x1^r and all its iterated derivatives. We show that when r = 2 the computation of a Gröbner basis of I with respect to the weighted lexicographical monomial order is related with an identity involving the partitions that appear in the first Rogers-Ramanujan identity. We then prove that a Gröbner basis of this ideal is not differentially finite in contrary with the case of the weighted reverse lexicographical order. We give a prove from this point of view of Gordon’s identities which is a family of important partitions identities. Using differential ideals we state a conjecture which could add a new member to Gordon’s identities. we prove then this conjecture for a special case. At the end, we give a simple and direct proof of a theorem of Nguyen Duc Tam about the Gröbner basis of the differential ideal [x1y1]; we then obtain identities involving partitions with 2 colors
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Ribeiro, Andreia Cristina. "Aspectos combinatorios de identidades do tipo Rogers-Ramanujan." [s.n.], 2006. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307502.

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Orientador: Jose Plinio de Oliveira Santos<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica<br>Made available in DSpace on 2018-08-07T19:25:43Z (GMT). No. of bitstreams: 1 Ribeiro_AndreiaCristina_D.pdf: 576297 bytes, checksum: 445154b7e26e801e909854c976d31c45 (MD5) Previous issue date: 2006<br>Resumo: Neste trabalho são estudadas varias das identidades do tipo Rogers-Ramanujan dadas por Slater. Em 1985, Andrews, introduziram um método geral para se estender para duas variáveis identidades desse tipo de modo a se obter, como casos especiais, certas importantes funções de Ramanujan. Santos, em 1991, forneceu conjecturas para varias das famílias de polinômios que surgem nestas extensões tendo provado algumas delas. Sills, em sua tese de doutorado, em 2002, implementou procedimentos que permitem a demonstra¸c¿ao das conjecturas dadas por Santos. No presente trabalho, de forma diferente daquela dada por Andrews, s¿ao introduzidos parâmetros nas somas que aparecem nestas identidades, de modo a se obter, em cada caso, funções geradoras que fornecem interpretações combinatórias para partições onde ¿números¿s¿ao vistos como ¿vetores¿e que fornecem, para especiais valores dos parâmetros, interpretações novas para muitas das identidades de Slater<br>Abstract: In this work many of the identities of the Rogers-Ramanujan type given by Slater are considered. In 1985, Andrews, introduced a general method in other to extend to two variables identities of this type in order to get, as special cases, some important functions of Ramanujan. Santos, in 1991, gave conjectures for many of the family of polynomials that appears in those extensions providing the proofs for some of them. Sills, in his Ph.D. thesis in 2002 ,has implemented procedures allowing the proofs of the conjectures given by Santos. In the present work, in a form different from the one given by Andrews, parameters are introduced in the sums of the identities in such a way to get, in each case, generating functions giving combinatorial interpretations for partitions where ¿numbers¿are represented as ¿vectors¿and that can give, as special cases, combinatorial interpretations for many of the identities given by Slater<br>Doutorado<br>Matematica Aplicada<br>Doutor em Matemática Aplicada
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Nyirenda, Darlison. "Analytic and combinatorial explorations of partitions associated with the Rogers-Ramanujan identities and partitions with initial repetitions." Thesis, 2016. http://hdl.handle.net/10539/21040.

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A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in ful lment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016.<br>In this thesis, various partition functions with respect to Rogers-Ramanujan identities and George Andrews' partitions with initial repetitions are studied. Agarwal and Goyal gave a three-way partition theoretic interpretation of the Rogers- Ramanujan identities. We generalise their result and establish certain connections with some work of Connor. Further combinatorial consequences and related partition identities are presented. Furthermore, we re ne one of the theorems of George Andrews on partitions with initial repetitions. In the same pursuit, we construct a non-diagram version of the Keith's bijection that not only proves the theorem, but also provides a clear proof of the re nement. Various directions in the spirit of partitions with initial repetitions are discussed and results enumerated. In one case, an identity of the Euler-Pentagonal type is presented and its analytic proof given.<br>M T 2016
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Books on the topic "Identités de Rogers-Ramanujan"

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An invitation to q-series: From Jacobi's triple product identity to Ramanujan's "most beautiful identity". World Scientific Pub Co., 2011.

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Alladi, Krishnaswami, Frank Garvan, and Ae Ja Yee. Ramanujan 125: International conference to commemorate the 125th anniversary of Ramanujan's birth, Ramanujan 125, November 5--7, 2012, University of Florida, Gainesville, Florida. American Mathematical Society, 2014.

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1939-, Berndt Bruce C., ed. Ramanujan's forty identities for the Rogers-Ramanujan functions. American Mathematical Society, 2007.

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Ramanujan's Forty Identities for the Rogers-ramanujan Functions (Memoirs of the American Mathematical Society). Amer Mathematical Society, 2007.

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Book chapters on the topic "Identités de Rogers-Ramanujan"

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Agarwal, A. K. "Rogers-Ramanujan Identities." In Current Trends in Number Theory. Hindustan Book Agency, 2002. http://dx.doi.org/10.1007/978-93-86279-09-5_2.

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Hirschhorn, Michael D. "The Rogers–Ramanujan Identities and the Rogers–Ramanujan Continued Fraction." In The Power of q. Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-57762-3_15.

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Andrews, George E., and Bruce Berndt. "Rogers-Ramanujan-Slater Type Identities." In Ramanujan’s Lost Notebook. Springer New York, 2005. http://dx.doi.org/10.1007/0-387-28124-x_12.

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Corteel, Sylvie, and Trevor Welsh. "The A2 Rogers–Ramanujan Identities Revisited." In Trends in Mathematics. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-57050-7_18.

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Andrews, George E. "An identity of sylvester and the rogers-ramanujan identities." In Lecture Notes in Mathematics. Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/bfb0086399.

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Lepowsky, James, and Robert Lee Wilson. "Z-Algebras and the Rogers-Ramanujan Identities." In Mathematical Sciences Research Institute Publications. Springer US, 1985. http://dx.doi.org/10.1007/978-1-4613-9550-8_6.

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Stanton, D. "Gaussian Integrals and the Rogers-Ramanujan Identities." In Developments in Mathematics. Springer US, 2001. http://dx.doi.org/10.1007/978-1-4613-0257-5_16.

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Bressoud, David M. "Lattice paths and the Rogers-Ramanujan identities." In Lecture Notes in Mathematics. Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/bfb0086403.

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Andrews, George E., and Bruce C. Berndt. "Ramanujan’s Forty Identities for the Rogers–Ramanujan Functions." In Ramanujan's Lost Notebook. Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-3810-6_8.

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Andrews, George E., and Bruce Berndt. "An Empirical Study of the Rogers-Ramanujan Identities." In Ramanujan’s Lost Notebook. Springer New York, 2005. http://dx.doi.org/10.1007/0-387-28124-x_11.

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Conference papers on the topic "Identités de Rogers-Ramanujan"

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Burge, William H. "Scratchpad and the Rogers-Ramanujan identities." In the 1991 international symposium. ACM Press, 1991. http://dx.doi.org/10.1145/120694.120722.

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