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Colloquium Home >>
UCCS MATHEMATICS
COLLOQUIUM
Thursday,
May 1, 2008
12:30 p.m.-1:30 p.m.
UC Room 307
(Refreshments at 12:15 pm)
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Parametric Representations for Eisenstein Series
from Ramanujan's Differential Equations
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Abstract: The parametric representations for Eisenstein series in terms of complete elliptic integrals arise in an enormous variety of contexts in number theory, and, in particular, in Ramanujan's work.
In this lecture I will derive these classical parameterizations without using the Jacobi-Ramanujan inversion formula relating theta functions and hypergeometric series. This approach, in contrast to the usual derivation, relies exclusively on the differential equations satisfied by certain Eisenstein series. One advantage of this technique is that it can be applied to derive additional parameterizations, including those involving Ramanujan's alternative bases. It may potentially be
applied to derive information about higher-order analogues of the classical theory in cases where no inversion formula is known.
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