On some continued fraction identities of Srinivasa Ramanujan

Bhargava, S.; Adiga, Chandrashekara

doi:10.1090/s0002-9939-1984-0749881-3

Cited by 20 publications

(11 citation statements)

References 6 publications

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“…A different continued fraction for the same primary quotient of q-series examined in Section 6.2 is studied in Section 6.4. In our discourse, we mainly follow the presentations by Andrews [26] and S. Bhargava and C. Adiga [91] in their papers. Some of our proofs of Ramanujan's corollaries depend on results of Andrews, L.J.…”

Section: Introductionmentioning

confidence: 99%

Ramanujan’s Lost Notebook

Andrews¹,

Berndt²

2005

227

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Section: Introductionmentioning

confidence: 99%

Ramanujan’s Lost Notebook

Andrews¹,

Berndt²

2005

227

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“…Closed-form expressions for continued fractions of this form are listed in Ramanujan's "lost" notebook (see the discussion in [5]), and evaluations for various ranges of the parameters (although not all the values we need) can be found in [5,32,33] (although in the last several parameter restrictions are omitted). …”

Section: Hitting Time Distributions For T Qmentioning

confidence: 99%

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Bhamidi¹,

Evans²,

Peled³

et al. 2008

Institute of Mathematical Statistics Collections

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Motivated by Lévy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be "continuous" in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator f → 1 2 f ′′ . We then consider the special case where the state space is the self-similar set {±q k : k ∈ Z} ∪ {0} for some q > 1. Using the scaling properties of the process, we represent the Laplace transforms of various hitting times as certain continued fractions that appear in Ramanujan's "lost" notebook and evaluate these continued fractions in terms of basic hypergeometric functions (that is, q-analogues of classical hypergeometric functions). The process has 0 as a regular instantaneous point, and hence its sample paths can be decomposed into a Poisson process of excursions from 0 using the associated continuous local time. Using the reversibility of the process with respect to the natural measure on the state space, we find the entrance laws of the corresponding Itô excursion measure and the Laplace exponent of the inverse local timeboth again in terms of basic hypergeometric functions. By combining these ingredients, we obtain explicit formulae for the resolvent of the process. We also compute the moments of the process in closed form. Some of our results involve q-analogues of classical distributions such as the Poisson distribution that have appeared elsewhere in the literature.

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“…20 The / =0,1,2 results can then be represented as infinite sums, based on the classical identities due to Ramanujan; see Bhargava and co-workers. 21,22 This leads to an Ansatz for general /,…”

Section: V-mmentioning

confidence: 99%

Exact generating function for fully directed compact lattice animals

Privman

Švrakić

1988

Phys. Rev. Lett.

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The fully directed compact lattice-animal model on the square lattice is solved exactly. An explicit expression is obtained for the cluster-number generating function which has a complicated complexplane singularity pattern including a natural boundary of essential singularities. However, the cluster numbers are controlled by a simple pole singularity and grow proportionally to X N , where TV is the number of sites, and X = 2.66.PACS numbers: 05.50.+q, 36.20.Ey, Models of lattice animals (connected clusters) have received much attention in recent years as simply systems with "geometrical" critical phenomena. Various aspects of the isotropic lattice animals have been reviewed, e.g.,

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On some continued fraction identities of Srinivasa Ramanujan

Abstract: Abstract. The main purpose of this note is to state and prove, in a simple, unified manner, several 17-continued fraction expansions found in Ramanujan's "lost" notebook. This is related to some recent works of G. E. Andrews and M. D. Hirschhorn.

Cited by 20 publications

References 6 publications

Ramanujan’s Lost Notebook

Ramanujan’s Lost Notebook

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Exact generating function for fully directed compact lattice animals

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