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Ramanujan’s unpublished manuscript on the partition and tau functions with proofs and commentary
 Sém. Lotharingien de Combinatoire 42 (1999), 63 pp.; in The Andrews Festschrift
, 2001
"... When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s taufunction τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising mat ..."
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Cited by 26 (13 self)
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When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s taufunction τ(n). The first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising material for insertion in the
Cubic Analogues Of The Jacobian Theta Function ...
, 1993
"... . There are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function 2 F 1 ( 1 3 ; 2 3 1 ; \Delta) analogous to the classical ` 2 (q), ` 3 (q), ` 4 (q) and the hypergeometric function 2 F 1 ( 1 2 ; 1 2 1 ; \Delta). We give elliptic function genera ..."
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Cited by 24 (13 self)
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. There are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function 2 F 1 ( 1 3 ; 2 3 1 ; \Delta) analogous to the classical ` 2 (q), ` 3 (q), ` 4 (q) and the hypergeometric function 2 F 1 ( 1 2 ; 1 2 1 ; \Delta). We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical thetafunction `(z; q). A number of identities are proved. The proofs are selfcontained, relying on nothing more than the Jacobi triple product identity. The second author completed this research while an NSERC International Fellow at Dalhousie. The third author was supported by NSERC of Canada. Received by the editors December 13, 1991. AMS Classification: 33D10. Secondary: 05A19, 11B65, 11F27, 33C05 c flCanadian Mathematical Society 1993. 1 x 1 Introduction and statement of results In a recent paper, Borwein, Borwein and Garvan [BBG] introduce three functions, a(q) = X q m 2 +mn+n 2 ; b(q) = X ! m\Gamman q m 2 +m...
Algebraic transformations of Gauss hypergeometric functions, preprint
, 2004
"... The paper classifies algebraic transformations of Gauss hypergeometric functions and pullback transformations between hypergeometric differential equations. This classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of th ..."
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Cited by 23 (9 self)
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The paper classifies algebraic transformations of Gauss hypergeometric functions and pullback transformations between hypergeometric differential equations. This classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function.
New 5F4 hypergeometric transformations, threevariable Mahler measures, and formulas for 1/π
 Ramanujan J
"... New relations are established between families of threevariable Mahler measures. Those identities are then expressed as transformations for the 5F4 hypergeometric function. We use these results to obtain two explicit 5F4 evaluations, and several new formulas for 1/π. MSC: 33C20, 33C05, 11F66 1 ..."
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Cited by 18 (7 self)
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New relations are established between families of threevariable Mahler measures. Those identities are then expressed as transformations for the 5F4 hypergeometric function. We use these results to obtain two explicit 5F4 evaluations, and several new formulas for 1/π. MSC: 33C20, 33C05, 11F66 1
Generalized elliptic integrals and modular equations
 Pacific J. Math
"... In geometric function theory, generalized elliptic integrals and functions arise from the SchwarzChristoffel transformation of the upper halfplane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic num ..."
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Cited by 17 (8 self)
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In geometric function theory, generalized elliptic integrals and functions arise from the SchwarzChristoffel transformation of the upper halfplane onto a parallelogram and are naturally related to Gaussian hypergeometric functions. Certain combinations of these integrals also occur in analytic number theory in the study of Ramanujan’s modular equations and approximations to π. The authors study the monotoneity and convexity properties of these quantities and obtain sharp inequalities for them. 1. Introduction. In 1995 B. Berndt, S. Bhargava, and F. Garvan published an important paper [BBG] in which they studied generalized modular equations and gave proofs for numerous statements concerning these equations made by Ramanujan in his unpublished notebooks. No record of Ramanujan’s original
Ramanujan’s series for 1/π: A survey
"... Beginning with Ramanujan’s epic paper, Modular Equations and Approximations to π, we describe Ramanujan’s series for 1/π and later attempts to prove them. Generalizations, analogues, and consequences of Ramanujan’s series are discussed. ..."
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Cited by 16 (2 self)
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Beginning with Ramanujan’s epic paper, Modular Equations and Approximations to π, we describe Ramanujan’s series for 1/π and later attempts to prove them. Generalizations, analogues, and consequences of Ramanujan’s series are discussed.
Ramanujan’s series for 1/π arising from his cubic and quartic theories of elliptic functions
 J. Math. Anal. Appl
"... Abstract. Using certain representations for Eisenstein series, we derive several of Ramanujan’s series for 1/π arising from his cubic and quartic theories of elliptic functions. ..."
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Cited by 10 (4 self)
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Abstract. Using certain representations for Eisenstein series, we derive several of Ramanujan’s series for 1/π arising from his cubic and quartic theories of elliptic functions.
Cubic modular equations and new Ramanujantype series for 1/π
 Pacific J. Math
"... In this paper, we derive new Ramanujantype series for 1/π which belong to “Ramanujan’s theory of elliptic functions to alternative base 3 ” developed recently by B.C. Berndt, S. Bhargava, and F.G. Garvan. 1. Introduction. Let (a)0 = 1 and, for a positive integer m, (a)m: = a(a + 1)(a +2)···(a+m−1), ..."
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Cited by 9 (2 self)
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In this paper, we derive new Ramanujantype series for 1/π which belong to “Ramanujan’s theory of elliptic functions to alternative base 3 ” developed recently by B.C. Berndt, S. Bhargava, and F.G. Garvan. 1. Introduction. Let (a)0 = 1 and, for a positive integer m, (a)m: = a(a + 1)(a +2)···(a+m−1), and ∞ ∑ (a)m(b)m z
Ramanujan's Explicit Values For The Classical ThetaFunction
"... this paper is to prove each of these new evaluations. We also establish some new evaluations of '(q) ..."
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Cited by 8 (7 self)
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this paper is to prove each of these new evaluations. We also establish some new evaluations of '(q)
Some Integrals Of Theta Functions In Ramanujan's Lost Notebook
 Proceedings of the Fifth Canadian Number Theory Association Meeting
"... . In his lost notebook, Ramanujan recorded certain identities involving integrals of theta functions. Some identities are proved by using identities relating theta functions and Lambert series. Two identities give integral representations for the RogersRamanujan continued fraction. 1. Introduction ..."
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Cited by 8 (2 self)
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. In his lost notebook, Ramanujan recorded certain identities involving integrals of theta functions. Some identities are proved by using identities relating theta functions and Lambert series. Two identities give integral representations for the RogersRamanujan continued fraction. 1. Introduction On pages 207 and 46 in his lost notebook [9], Ramanujan recorded eight identities involving integrals of theta functions. Two of these give integral representations for the RogersRamanujan continued fraction, defined by F (q) := 1 1 + q 1 + q 2 1 + q 3 1 + \Delta \Delta \Delta jqj ! 1: More precisely, on page 46, Ramanujan claims that q 1=5 F (q) = p 5 \Gamma 1 2 exp ` \Gamma 1 5 Z 1 q (1 \Gamma t) 5 (1 \Gamma t 2 ) 5 \Delta \Delta \Delta (1 \Gamma t 5 )(1 \Gamma t 10 ) \Delta \Delta \Delta dt t ' = p 5 \Gamma 1 2 \Gamma p 5 1 + 3 + p 5 2 exp ` 1 p 5 Z q 0 (1 \Gamma t) 5 (1 \Gamma t 2 ) 5 \Delta \Delta \Delta (1 \Gamma t 1=5 )(1 \Gamma t ...