Browsing by Subject "Modular equations"

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Open Access
Elementary proofs of some identities of Ramanujan for the Rogers-Ramanujan functions
(Elsevier, 2012) Yesilyurt, H.
In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. With one exception all of Ramanujan's identities were proved. In this paper, we provide a proof for the remaining identity together with new elementary proofs for two identities of Ramanujan which were previously proved using the theory of modular forms. Ramanujan stated that each of his formula was the simplest of a large class. Our proofs are constructive and permit us to obtain several analogous identities which could have been stated by Ramanujan and may very well belong to his large class of identities. © 2011 Elsevier Inc.
Open Access
Modular equations of degrees 13, 29, and 61
(Springer New York LLC, 2023-11-21) Güloğlu, Ahmet M.; Yesilyurt, Hamza
Schröter-type theta function identities were very instrumental in proving modular equations. In this paper, by employing a generalization of this identity, we prove for the ﬁrst time a modular equation of degree 61. Furthermore, new modular equations of degrees 13 and 29 are obtained.
Open Access
New identities for 7-cores with prescribed BG-rank
(Elsevier BV * North-Holland, 2008) Berkovich, A.; Yesilyurt, H.
Let π be a partition. BG-rank(π) is defined as an alternating sum of parities of parts of π [A. Berkovich, F.G. Garvan, On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo 5 and generalizations, Trans. Amer. Math. Soc. 358 (2006) 703-726. [1]]. Berkovich and Garvan [The BG-rank of a partition and its applications, Adv. in Appl. Math., to appear in 〈http://arxiv.org/abs/math/0602362〉] found theta series representations for the t-core generating functions ∑n ≥ 0 at, j (n) qn, where at, j (n) denotes the number of t-cores of n with BG-rank = j. In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme values of BG-rank. In this paper we discuss representations of this type for all 7-cores with prescribed BG-rank. We make an essential use of the Ramanujan modular equations of degree seven [B.C. Berndt, Ramanujan's Notebooks, Part III, Springer, New York, 1991] to prove a variety of new formulas for the 7-core generating functionunder(∏, j ≥ 1) frac((1 - q7 j)7, (1 - qj)) .These formulas enable us to establish a number of striking inequalities for a7, j (n) with j = - 1, 0, 1, 2 and a7 (n), such asa7 (2 n + 2) ≥ 2 a7 (n), a7 (4 n + 6) ≥ 10 a7 (n) . Here a7 (n) denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only. © 2007 Elsevier B.V. All rights reserved.
Open Access
On the representations of integers by the sextenary quadratic form x2 + y2 + z2 + 7 s2 + 7 t2 + 7 u2 and 7-cores
(Academic Press, 2009) Berkovich, A.; Yesilyurt H.
In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x2 + y2 + z2 + 7 s2 + 7 t2 + 7 u2. We establish the following intriguing inequalities2 ω (n + 2) ≥ a7 (n) ≥ ω (n + 2) for n ≠ 0, 2, 6, 16 . Here a7 (n) is the number of partitions of n that are 7-cores and ω (n) is the number of representations of n by the sextenary form (x2 + y2 + z2 + 7 s2 + 7 t2 + 7 u2) / 8 with x, y, z, s, t and u being odd positive integers. © 2008 Elsevier Inc. All rights reserved.