QUINTUPLE PRODUCT IDENTITY AS A SPECIAL CASE OF RAMANUJAN'S 1ψ1 SUMMATION FORMULA

In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1ψ1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q -binomial theorem.

Euler's Pentagonal Number Theorem Implies the Jacobi Triple Product Identity

By means of Liouville's theorem, we show that Euler's pentagonal number theorem implies the Jacobi triple product identity.

Two Simple Proofs of Winquist's Identity

We give two new proofs of Winquist's identity. In the first proof, we use basic properties of cube roots of unity and the Jacobi triple product identity. The latter does not use the Jacobi triple product identity.

CONSTRUCTION OF EISENSTEIN SERIES FOR Γ0(p)

A simple construction of Eisenstein series for the congruence subgroup Γ0(p) is given. The construction makes use of the Jacobi triple product identity and Gauss sums, but does not use the modular transformation for the Dedekind eta-function. All positive integral weights are handled in the same way, and the conditionally convergent cases of weights 1 and 2 present no extra difficulty.

Cubic Analogues of the Jacobian Theta Function θ(z, q)

There are three modular forms a(q), b(q), c(q) involved in the parametrization of the hypergeometric function analogous to the classical θ2(q), θ3(q), θ4(q) and the hypergeometric function We give elliptic function generalizations of a(q), b(q), c(q) analogous to the classical theta-function θ(z, q). A number of identities are proved. The proofs are self-contained, relying on nothing more than the Jacobi triple product identity