References & Citations
Mathematics > Classical Analysis and ODEs
Title: Miscellaneous summation, integration, and transformation formulas
(Submitted on 16 Apr 2024)
Abstract: 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.
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