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Showing 1 to 20 of 47 for “"Ramanujan"”.
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On the Rogers-Ramanujan and Ramanujan-Gollnitz-Gordon Continued Fractions
In a manuscript of Ramanujan, published with his Lost Notebook, there are forty identities involving the Rogers-Ramanujan functions. According to G. N. Watson, the beauty of these identities is comparable to that of the Rogers-Ramanujan identities. In the final chapter, we establish modular …
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Some important continued fractions of Ramanujan and Selberg
… for three q-difference equations which arise in Ramanujan and Selberg's work on q-continued fractions. From these solutions, we derive criteria for the convergence of three Ramanujan-Selberg continued fractions when q is a primitive m-th root of unity. Moreover, when the continued fractions …
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Bounds on k-Regular Ramanujan Graphs and Separator Theorems
… type of expander graph in a. certain sense is a Ramanujan graph. Families of graphs that have separator theorems fail to be Ramanujan if the vertex set gets sufficiently large. Using separator theorems to get an estimate on the expanding constant of graphs, we get bounds 011 the number of …
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Modular identities for the Rogers-Ramanujan functions and analogues
Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by William Ingram (wingram2@illinois.edu) on 2011-01-21T22:47:37Z Item is restricted until 2013-01-21T22:47:37Z
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The Expanding Constant, Ramanujan Graphs, and Winnie Li Graphs
… and for the proof that Winnie Li's graphs are Ramanujan. The paper also establishes the implications of the Ramanujan property for the expanding constant.
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Eisenstein Series, Analogues of the Rogers -Ramanujan Functions, and Partition Identities
… inspired by the work of O. Kolberg and S. Ramanujan. We use functions studied by N. J. Fine and R. J. Evans to construct analogues of modular equations, and then derive new identities satisfied by n=0infinity p(ln + k) qn, where p(n) is the ordinary partition function, l is an odd prime, …
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Zeros of Generalized Rogers -Ramanujan Series and Topics From Ramanujan's Theory of Elliptic Functions
… equations analogous to relations given by Ramanujan in Chapter 21 of his Second Notebook. We also use trigonometric interpolation to write many of Ramanujan's identities in terms of the Weierstrass ℘-function.
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Contributions to Ramanujan's Theories of Modular Equations, Ramanujan-Type Series for 1/pi, and Partitions
Embargo set by: Seth Robbins for item 88266 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs
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The Rogers -Ramanujan Continued Fraction and a Certain Quotient of ETA Functions Found in Ramanujan's Lost Notebook
… which is essential for an elementary proof for Ramanujan's famous partition identity modulo 11, p(11n + 6) ≡ 0 (mod 11) is provided in the last part of the thesis.
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Contributions to the Theory of Q-Series and Mock Theta Functions
… time, a series of four related identities from Ramanujan's lost notebook. These are one-parameter identities which would directly imply some of the relations for the third order mock theta functions given by Ramanujan. In Chapter 3, we prove very general theorems on the periodicity of signs of …
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Congruences in modular, Jacobi, Siegel, and mock modular forms with applications
… of modular and other automorphic forms. Ramanujan famously found congruences for the partition function like p(5n+4) = 0 mod 5. For a wide class of modular forms, we classify the primes for which there can be analogous congruences in the coefficients of the Fourier expansion. We have …
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On the Convergence and Divergence of Q-Continued Fractions on and Off the Unit Circle
… in a certain class, which includes the Rogers-Ramanujan continued fractions and the three ""Ramanujan-Selberg"" continued fractions, diverge in the general sense at an uncountable set of points on the unit circle. We also show that the Rogers-Ramanujan continued fraction converges generally at …
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Generalizations of Certain Results on Continued Fraction
… thesis we study generalizations of the Rogers-Ramanujan continued fraction. The Rogers-Ramanujan continued fraction arises from a three-term q-difference equation. We consider (m + 1)-term q-difference equations and also a generalization of the continued fraction algorithm called a G-continued …
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Modular equations and Ramanujan's cubic and quartic theories of theta functions
… thesis, we prove several identities involving Ramanujan's general theta function. In Chapter 2, we give proofs for new Ramanujan type modular equations discovered by Somos and establish applications of some of them. In Chapter 3, we will give proofs for several Dedekind eta product identities …
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Unique Games Conjecture : the Boolean Hypercube and connections to graph lifts
… for studying lifts has been understanding Ramanujan expander graphs via two key questions: Is a ``typical'' lift of an expander graph also an expander; and how can we (efficiently) construct Ramanujan expanders using lifts? In our work we continue the study of Graph Lifts and show that, for …
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Partition identity bijections related to sign-balance and rank
… and Andrews' generalization of the Rogers-Ramanujan identities. This gives a new combinatorial proof of the first Rogers-Ramanujan identity. We also relate this work to Garvan's generalization of rank. In the second part, we prove a family of four-parameter partition identities which …
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Q-Continued Fractions and Related Q-Series
… 5, we prove two other equations on page 45 of Ramanujan's lost notebook by using the Bauer-Muir transformation and by using the method of successive approximation. Also, we prove several continued fractions of Ramanujan which are related to the odd parts of continued fractions.
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Contributions to Ramanujan's continued fractions, class invariants, partition identities and modular equations
Various topics related to the work of Ramanujan are discussed in this thesis. In Chapter 2, we give a new proof of Ramanujan's famous partition identity modulo 5 (see (1.1)). This proof is an improvement of W. N. Bailey's proof given in 1952. We also establish a new proof of Ramanujan's partition …
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Transformation formulas associated with integrals involving the Riemann Ξ-function and rank-crank type PDEs in partition theory
… found on page 220 in the volume ``Ramanujan's lost notebook and other unpublished papers''. In Chapter 2, we give two different proofs of this result, the second one being implicitly sought by A.P.~Guinand while rediscovering this result. However, Guinand only partially rediscovered …
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Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi
We introduce classes of Ramanujan-like series for $\frac{1}{\pi}$, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting $H_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ denote the …
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