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 ∑ₙ cₙeπ i γ n²; 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 ∑n≥ 1χ(n)/n\,φ(n) with Dirichlet characters; Trigonometric Fourier series expansion of hypergeometric functions of the argument sin²x; Modifications of the inverse tangent integral and identities for corresponding infinite products.