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Miscellaneous summation, integration, and transformation formulas

Martin Nicholson

TL;DR

This work presents a collection of Fourier-analytic formulas connecting summation, integration, and transformation techniques with special functions and $q$-series. It develops quadratic-exponential series reductions via Poisson summation, constructs a Fourier-Gauss transform framework with fusion of integrals and eigenfunctions of the cosine transform, and derives Dirichlet-character weighted summation identities tied to Dirichlet $L$-values. It also explores trigonometric Fourier series representations for hypergeometric functions of the argument $\sin^2 x$, including quadratic-transform and Appell–Lerch connections, and culminates with inverse tangent integrals and associated infinite products expressed through dilogarithms and generalized trigonometric constructs. Collectively, the paper reveals deep interplays between modular-type transformations, $q$-series fusion, and special-function representations with explicit closed forms and integral expressions.

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 $\sum_{n\ge 1}\frac{χ(n)}{n}\,\varphi(n)$ 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.

Miscellaneous summation, integration, and transformation formulas

TL;DR

This work presents a collection of Fourier-analytic formulas connecting summation, integration, and transformation techniques with special functions and -series. It develops quadratic-exponential series reductions via Poisson summation, constructs a Fourier-Gauss transform framework with fusion of integrals and eigenfunctions of the cosine transform, and derives Dirichlet-character weighted summation identities tied to Dirichlet -values. It also explores trigonometric Fourier series representations for hypergeometric functions of the argument , including quadratic-transform and Appell–Lerch connections, and culminates with inverse tangent integrals and associated infinite products expressed through dilogarithms and generalized trigonometric constructs. Collectively, the paper reveals deep interplays between modular-type transformations, -series fusion, and special-function representations with explicit closed forms and integral expressions.

Abstract

This is a discussion of miscellaneous summation, integration and transformation formulas obtained using Fourier analysis. The topics covered are: Series of the form ; Fusion of integrals, and in particular fusion of -beta integrals related to Gauss-Fourier transform, and a related family of eigenfunctions of the cosine Fourier transform; Summation formulas of the type with Dirichlet characters; Trigonometric Fourier series expansion of hypergeometric functions of the argument ; Modifications of the inverse tangent integral and identities for corresponding infinite products.
Paper Structure (5 sections, 13 theorems, 153 equations)

This paper contains 5 sections, 13 theorems, 153 equations.

Table of Contents

  1. Series with general term cnexp(piign2)
  2. Fourier-Gauss transform and fusion of integrals
  3. Series with general term chi(n)f(an)/n
  4. Trigonometric series for hypergeometric functions
  5. Inverse tangent integral and associated infinite products

Key Result

Proposition 1

Let $k\in\mathbb{R}$. Let $u_n(k)$, $v_n(k)$ be absolutely summable sequences. For $x\in\mathbb{R}$ and $u_n(k)$ define $\phi_u(x,k)$, $\psi_u(x,k)$ by phi, psi and similarly $\phi_v(x,k)$, $\psi_v(x,k)$ for $v_n(k)$. Then for $m\in\mathbb{R}$

Theorems & Definitions (13)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Lemma 8
  • Proposition 9
  • Proposition 10
  • ...and 3 more