Miscellaneous summation, integration, and transformation formulas

This is a discussion of miscellaneous summation, integration and transformation formulas obtained using Fourier analysis. The topics covered are: Series of the form ${\sum }_{n\in \mathbb{Z}}{c}_n{e}^{\pi i\gamma n^2}$; 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 ${\sum }_{n\ge 1}\frac{\chi (n)}{n}\phi (n)$ with Dirichlet characters; Trigonometric Fourier series expansion of hypergeometric functions of the argument ${\sin }^{2}x$; Modifications of the inverse tangent integral and identities for corresponding infinite products.