Some applications to integer partitions are given. We also prove another addition theorem for the q-exponential function. The addition theorem and the q-plane wave formula are used to evaluate some definite integrals and establish certain power series identities. The resulting identities give new proofs and generalizations of the Rogers-Ramnujan identities. A quintic transformation which immediately proves the Rogers-Ramnujan identities without the Jacobi triple product identity is also given.
Similar techniques lead to new transformations for unilateral and bilateral series.
Iterates of these versions give many new versions of multisum Rogers-Ramanujan identities. We also prove Melzer's conjectures for the Fermionic forms of the supersymmetric analogues of Virasoro characters. An introduction is given to the use orthogonal polynomials in distance regular graphs and enumeration. Some examples in each area are given, along with open problems. It is well known that the Fourier transform of a Gaussian is Gaussian.