97 research outputs found
On Plouffe's Ramanujan Identities
Recently, Simon Plouffe has discovered a number of identities for the Riemann
zeta function at odd integer values. These identities are obtained numerically
and are inspired by a prototypical series for Apery's constant given by
Ramanujan: Such sums follow from a general relation given by Ramanujan, which is
rediscovered and proved here using complex analytic techniques. The general
relation is used to derive many of Plouffe's identities as corollaries. The
resemblance of the general relation to the structure of theta functions and
modular forms is briefly sketched.Comment: 19 pages, 3 figures; v4: minor corrections; modified intro; revised
concluding statement
Uncovering Ramanujan's "Lost" Notebook: An Oral History
Here we weave together interviews conducted by the author with three
prominent figures in the world of Ramanujan's mathematics, George Andrews,
Bruce Berndt and Ken Ono. The article describes Andrews's discovery of the
"lost" notebook, Andrews and Berndt's effort of proving and editing Ramanujan's
notes, and recent breakthroughs by Ono and others carrying certain important
aspects of the Indian mathematician's work into the future. Also presented are
historical details related to Ramanujan and his mathematics, perspectives on
the impact of his work in contemporary mathematics, and a number of interesting
personal anecdotes from Andrews, Berndt and Ono
Ramanujan and Extensions and Contractions of Continued Fractions
If a continued fraction is known to converge
but its limit is not easy to determine, it may be easier to use an extension of
to find the limit. By an extension of
we mean a continued fraction whose odd or even part is . One can
then possibly find the limit in one of three ways:
(i) Prove the extension converges and find its limit;
(ii) Prove the extension converges and find the limit of the other
contraction (for example, the odd part, if is the
even part);
(ii) Find the limit of the other contraction and show that the odd and even
parts of the extension tend to the same limit.
We apply these ideas to derive new proofs of certain continued fraction
identities of Ramanujan and to prove a generalization of an identity involving
the Rogers-Ramanujan continued fraction, which was conjectured by Blecksmith
and Brillhart.Comment: 16 page
On an Asymptotic Series of Ramanujan
An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is
concerned with the behavior of the expected value of for large
where is a Poisson random variable with mean and
is a function satisfying certain growth conditions. We generalize this by
studying the asymptotics of the expected value of when the
distribution of belongs to a suitable family indexed by a convolution
parameter. Examples include the problem of inverse moments for distribution
families such as the binomial or the negative binomial.Comment: To appear, Ramanujan
q-Newton binomial: from Euler to Gauss
A counter-intuitive result of Gauss (formulae (1.6), (1.7) below) is made
less mysterious by virtue of being generalized through the introduction of an
additional parameter
Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms
A unified treatment is given of low-weight modular forms on \Gamma_0(N),
N=2,3,4, that have Eisenstein series representations. For each N, certain
weight-1 forms are shown to satisfy a coupled system of nonlinear differential
equations, which yields a single nonlinear third-order equation, called a
generalized Chazy equation. As byproducts, a table of divisor function and
theta identities is generated by means of q-expansions, and a transformation
law under \Gamma_0(4) for the second complete elliptic integral is derived.
More generally, it is shown how Picard-Fuchs equations of triangle subgroups of
PSL(2,R) which are hypergeometric equations, yield systems of nonlinear
equations for weight-1 forms, and generalized Chazy equations. Each triangle
group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic
Two-divisibility of the coefficients of certain weakly holomorphic modular forms
We study a canonical basis for spaces of weakly holomorphic modular forms of
weights 12, 16, 18, 20, 22, and 26 on the full modular group. We prove a
relation between the Fourier coefficients of modular forms in this canonical
basis and a generalized Ramanujan tau-function, and use this to prove that
these Fourier coefficients are often highly divisible by 2.Comment: Corrected typos. To appear in the Ramanujan Journa
Exclusion statistics: A resolution of the problem of negative weights
We give a formulation of the single particle occupation probabilities for a
system of identical particles obeying fractional exclusion statistics of
Haldane. We first derive a set of constraints using an exactly solvable model
which describes an ideal exclusion statistics system and deduce the general
counting rules for occupancy of states obeyed by these particles. We show that
the problem of negative probabilities may be avoided with these new counting
rules.Comment: REVTEX 3.0, 14 page
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