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functional identities that, apart from being amazingly amusing in themselves, find application in the derivation of Ramanujan-type formulas for
and in the computation of mathematical constants.
We employ a modular method to establish the new result that two types of Eisenstein series to the tredecic base may be parametrised in terms of the eta quotients
. The method can also be used to give short and simple proofs for the analogous cubic, quintic and septic theories.
A general theorem is stated that unifies 93 rational Ramanujan-type series for 1/π, 40 of which are believed to be new. Moreover, each series is shown to have a companion identity, thereby giving another 93 series, the majority of which are new.
Generating functions are used to derive formulas for the number of representations of a positive integer by each of the quadratic forms x12+x22+x32+2x42, x12+2x22+2x32+2x42, x12+x22+2x32+4x42 and x12+2x22+4x32+4x42. The formulas show that the number of representations by each form is always positive. Some of the analogous results involving sums of triangular numbers are also given.
We prove an observation associated with η3(τ)η3(7τ) which is found on page 54 of Ramanujan’s Lost Notebook (S. Ramanujan, The Lost Notebook and Other Unpublished Papers (Narosa, New Delhi, 1988)). We then study functions of the type η3(aτ)η3(bτ) with a+b=8.
We give a simple proof of the identity
The proof uses only a few well-known properties of the cubic theta functions a(q), b(q) and c(q). We show this identity implies the interesting definite integral
A new, elementary proof of the Macdonald identities for An−1 using induction on n is given. Specifically, the Macdonald identity for An is deduced by multiplying the Macdonald identity for An−1 and n Jacobi triple product identities together.
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