  This is a discussion of miscellaneous summation, integration and
transformation formulas obtained using Fourier analysis. The topics covered
are: Series of the form $\sum_{n\in\mathbb{Z}} c_ne^{\pi i \gamma n^2}$; Fusion
of integrals, and in particular fusion of $q$-beta integrals related to
Gauss-Fourier transform, and a related family of eigenfunctions of the cosine
Fourier transform; Summation formulas of the type $\sum_{n\ge
1}\frac{\chi(n)}{n}\,\varphi(n)$ with Dirichlet characters; Trigonometric
Fourier series expansion of hypergeometric functions of the argument $\sin^2x$;
Modifications of the inverse tangent integral and identities for corresponding
infinite products.
