Browsing by Author "Berkovich, A."
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Item 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.Item Open Access On Rogers-Ramanujan functions, binary quadratic froms and eta-quotients(American Mathematical Society, 2014) Berkovich, A.; Yeşilyurt, H.In a handwritten manuscript published with his lost notebook, Ramanujan stated without proofs forty identities for the Rogers-Ramanujan functions. We observe that the function that appears in Ramanujan's identities can be obtained from a Hecke action on a certain family of eta products. We establish further Hecke-type relations for these functions involving binary quadratic forms. Our observations enable us to find new identities for the Rogers-Ramanujan functions and also to use such identities in return to find identities involving binary quadratic forms. © 2013 American Mathematical Society.Item 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.Item Open Access Ramanujan ' s identities and representation of integers by certain binary and quaternary quadratic forms(Springer New York LLC, 2009) Berkovich, A.; Yesilyurt, H.We revisit old conjectures of Fermat and Euler regarding the representation of integers by binary quadratic form x2+5y2. Making use of Ramanujan's 1ψ1 summation formula, we establish a new Lambert series identity for Σ∞n,m=-∞qn2+5m2 Conjectures of Fermat and Euler are shown to follow easily from this new formula. But we do not stop there. Employing various formulas found in Ramanujan's notebooks and using a bit of ingenuity, we obtain a collection of new Lambert series for certain infinite products associated with quadratic forms such as x2+6y2, 2x2+3y2, x2+15y2, 3x2+5y2, x2+27y2, x2+5(y2+z2+w2), 5x2+y2+z2+w2. In the process, we find many new multiplicative eta-quotients and determine their coefficients. © 2009 Springer Science+Business Media, LLC.Item Open Access Ramanujan's circular summation, t-cores and twisted partition identities(Elsevier, 2019) Berkovich, A.; Garvan, F.; Yeşilyurt, HamzaIn this paper, we give new evaluations for Ramanujan’s circular summation function. We also provide simpler proofs for known evaluations and give some generalizations. We discover modular relations among circular summation function partition function and give uniform proof of Ramanujan’s partition congruences for the moduli 5, 7 and 11. We also prove several interesting congruence relations.