English

Two Definite Integrals Involving Products of Four Legendre Functions

Classical Analysis and ODEs 2018-01-26 v2 Number Theory
Authors: Yajun Zhou
View on arXiv ↗ PDF ↗ DOI ↗

Abstract

The definite integrals ∫−11x[Pν(x)]4 dx \int_{-1}^1x[P_\nu(x)]^4\,\mathrm{d} x and ∫01x[Pν(x)]2{[Pν(x)]2−[Pν(−x)]2} dx \int_{0}^1x[P_\nu(x)]^2\{[P_\nu(x)]^2-[P_\nu(-x)]^2\}\,\mathrm{d} x are evaluated in closed form, where Pν P_\nu stands for the Legendre function of degree ν∈C \nu\in\mathbb C. Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green's functions and Epstein zeta functions.

Keywords

definite integrals Legendre functions closed-form evaluation automorphic Green's functions Epstein zeta functions products of four Legendre functions

Cite

@article{arxiv.1603.03547,
  title  = {Two Definite Integrals Involving Products of Four Legendre Functions},
  author = {Yajun Zhou},
  journal= {arXiv preprint arXiv:1603.03547},
  year   = {2018}
}

Comments

12 pages, accepted version. Proof of two integral formulae stated in arXiv:1301.1735v4 and arXiv:1506.00318v2

Related papers

View all related →
Classical Analysis and ODEs · Mathematics
On Some Integrals Over the Product of Three Legendre Functions
Yajun Zhou
2014-10-30
Classical Analysis and ODEs · Mathematics
Legendre Functions, Spherical Rotations, and Multiple Elliptic Integrals
Yajun Zhou
2014-07-21
Classical Analysis and ODEs · Mathematics
Some integrals and series involving the Gegenbauer polynomials and the Legendre functions on the cut (-1,1)
Radosław Szmytkowski
2011-11-15
Classical Analysis and ODEs · Mathematics
Definite Integrals using Orthogonality and Integral Transforms
Howard S. Cohl, Hans Volkmer
2012-10-22
Classical Analysis and ODEs · Mathematics
The Integral Representation for the Product of Two Parabolic Cylinder Functions $D_\nu (x) D_\nu (-x)$ at $Re \nu <0$ by Means of the Fundamental Solution of a Landau-Type Operator
C. Malyshev
2018-09-17
Classical Analysis and ODEs · Mathematics
On the derivatives $\partial^{2}P_{\nu}(z)/\partial\nu^{2}$ and $\partial Q_{\nu}(z)/\partial\nu$ of the Legendre functions with respect to their degrees
Radosław Szmytkowski
2017-07-11
Mathematical Physics · Physics
General integral-identities involving Legendre polynomials and their derivatives
Yannick Wunderlich, Kyungseon Joo, Victor I. Mokeev
2025-09-30
Classical Analysis and ODEs · Mathematics
A Remarkable Definite Integral
M. L. Glasser
2013-09-17
Classical Analysis and ODEs · Mathematics
A simple derivation of the integrals of products of Legendre polynomials with logarithmic weight
Sebastian Schmutzhard-Höfler
2024-04-18
Classical Analysis and ODEs · Mathematics
Discrete Mehler-Fock transforms
Semyon Yakubovich
2019-10-21
Classical Analysis and ODEs · Mathematics
Remarks on the Zeros of the Associated Legendre Functions with Integral Degree
J. F. van Diejen
2010-09-27
Classical Analysis and ODEs · Mathematics
On Parameter Differentiation for Integral Representations of Associated Legendre Functions
Howard S. Cohl
2015-03-17
Classical Analysis and ODEs · Mathematics
Integrals of products of four modified Bessel functions
Robert E. Gaunt
2026-01-21
General Mathematics · Mathematics
Definite integrals involving combinations of powers and logarithmic functions of complicated arguments expressed in terms of the Hurwitz zeta function
Robert Reynolds, Allan Stauffer
2021-04-30
Classical Analysis and ODEs · Mathematics
A note on a generalisation of a definite integral involving the Bessel function of the first kind
S A Dar, M Kamarujjama, R B Paris
2022-05-09
General Mathematics · Mathematics
A Sextuple Integral Containing the Product of Associated Legendre polynomials $P_v^u(x) P_{\nu }^{\mu }(y)$: Derivation and Evaluation
Robert Reynolds, Allan Stauffer
2022-01-11
Mathematical Physics · Physics
A family of Exponential Integrals suggested by Stellar Dynamics
Luca Ciotti
2020-09-15
General Mathematics · Mathematics
A collection of integrals, products and series
Robert Reynolds
2024-03-01
Classical Analysis and ODEs · Mathematics
On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order
Radoslaw Szmytkowski
2008-03-28
Complex Variables · Mathematics
Positive-definiteness and integral representations for special functions
Jorge Buescu, António Paixão
2018-01-30
Classical Analysis and ODEs · Mathematics
Legendre Functions of Fractional Degree: Transformations and Evaluations
Robert S. Maier
2018-06-22
Classical Analysis and ODEs · Mathematics
On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order)
Radoslaw Szmytkowski
2009-10-26
Mathematical Physics · Physics
A Generalisation For The Infinite Integral Over Three Spherical Bessel Functions
R. Mehrem, A. Hohenegger
2011-08-29
Classical Analysis and ODEs · Mathematics
Two Definite Integrals That Are Definitely (and Surprisingly!) Equal
Shalosh B. Ekhad, Doron Zeilberger, Wadim Zudilin
2020-09-24
Classical Analysis and ODEs · Mathematics
Berndt-type Integrals: Unveiling Connections with Barnes Zeta and Jacobi Elliptic Functions
Zachary P. Bradshaw, Christophe Vignat
2025-02-23