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arXiv · 2404.10805

Miscellaneous summation, integration, and transformation formulas

Martin Nicholson
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Abstract

This is a discussion of miscellaneous summation, integration and transformation formulas obtained using Fourier analysis. The topics covered are: Series of the form $\sum_{n\in\mathbb{Z}} c_ne^{πi γn^2}$; 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 $\sum_{n\ge 1}\frac{χ(n)}{n}\,φ(n)$ 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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Martin Nicholson. 2025-02-11. Miscellaneous summation, integration, and transformation formulas. https://arxiv.org/abs/2404.10805

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