290 research outputs found
Certain Integrals Arising from Ramanujan's Notebooks
In his third notebook, Ramanujan claims that In a following cryptic line, which
only became visible in a recent reproduction of Ramanujan's notebooks,
Ramanujan indicates that a similar relation exists if were replaced by
in the first integral and were inserted in the integrand of
the second integral. One of the goals of the present paper is to prove this
claim by contour integration. We further establish general theorems similarly
relating large classes of infinite integrals and illustrate these by several
examples
Two-dimensional series evaluations via the elliptic functions of Ramanujan and Jacobi
We evaluate in closed form, for the first time, certain classes of double
series, which are remindful of lattice sums. Elliptic functions, singular
moduli, class invariants, and the Rogers--Ramanujan continued fraction play
central roles in our evaluationsComment: 12 page
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
The reckoning of certain quartic and octic Gauss sums
In this brief note, we evaluate certain quartic and octic Gauss sums with the use of theorems on fourth and eighth power difference sets. We recall that a subset H of a finite (additive) abelian group G is said to be a difference set of G [5, p. 64] if for some fixed natural number λ, every nonzero element of G can be written as a difference of two elements of H fin exactly λ ways
A multi-variable theta product
AbstractA multi-variable theta product is examined. It is shown that, under very general choices of the parameters, the quotient of two such general theta products is a root of unity. Special cases are explicitly determined. The second main theorem yields an explicit evaluation of a sum of series of cosines; which greatly generalizes one of Ramanujan's theorems on certain sums of hyperbolic cosines
A fragment on Euler\u27s constant in Ramanujan\u27s lost notebook
A formula for Euler’s constant found in Ramanujan’s lost notebook and also in a problem he submitted to the Journal of the Indian Mathematical Society is proved and discussed
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