Legendre equation

The differential equation \((1-x^2)y''-2xy'+\nu(\nu+1)y=0\), whose solutions include Legendre polynomials for nonnegative integer \(\nu\) and Legendre functions more generally.

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Legendre polynomials: A sequence of orthogonal polynomials solving the Legendre equation when its degree is a nonnegative integer. They are the polynomial solutions selected by integer degree.
Linear ordinary differential equation: A differential equation in one independent variable whose dependent variable and derivatives occur linearly. The Legendre equation is a second-order member of this broad class.
Laplace's equation: The partial differential equation stating that a function’s Laplacian is zero. Separation of variables in spherical coordinates produces the Legendre equation for axisymmetric solutions.
Associated Legendre equation: A differential equation extending the Legendre equation with an additional order parameter m. It governs non-axisymmetric spherical modes and introduces the order absent here.
Legendre functions: Solutions of the Legendre equation defined for general degrees and orders, including functions of the first and second kinds. They extend the equation’s solutions beyond polynomial degrees.
Legendre differential operator: The operator \(\frac{d}{dx}[(1-x^2)\frac{d}{dx}]\), whose eigenfunctions are Legendre functions. It isolates the derivative terms of the equation as an eigenvalue problem.
Spherical harmonics: An orthonormal basis of angular functions on the sphere, built from associated Legendre functions and complex exponentials. Their polar-angle factors solve the associated Legendre equation.
Chebyshev differential equation: A differential equation whose polynomial solutions are Chebyshev polynomials of the first kind. Its polynomial eigenfunctions use a different weight and endpoint behavior.
Regular singular point: A singular point of a linear differential equation where Frobenius-series solutions can be constructed. The equation has regular singular points at both endpoints, x = ±1.
Eigenvalue problem: A problem seeking nonzero functions that an operator maps to scalar multiples of themselves. The parameter ν(ν+1) acts as the eigenvalue for the Legendre operator.