In mathematics, Legendre functions are solutions to Legendre's differential equation:

Upon seeing some samples of the straw from which the Leghorn hats were

then the orthogonal polynomial can be represented by a Rodrigues formula:

where Pn is the Legendre polynomials, which is equivalent to

The Legendre polynomials can be defined as the coefficients in the expansion of the generating function

is essentially the associated Legendre polynomial P ℓ m ( cos ⁡ θ ) {\displaystyle P_{\ell }^{m}(\cos \theta )}

This technique uses the orthogonality relation of the Legendre polynomials to expand an image as a double sum of those functions.

n and their special cases Legendre polynomials, Chebyshev polynomials, Gegenbauer polynomials can be written in terms of hypergeometric functions using

For example he published An unusual type of expansion problem (1924), A comparison of the series of Fourier and Birkhoff (1926), Developments in Legendre polynomials (1926), and Developments in Hermite polynomials (1927).

If we normalize the polynomials of Legendre, the coefficients of Fourier of the reducer φ related to this orthonormal system are null for an even index and are given by

He also wrote two papers which were published in 1950, the first being On a simple property of the derivatives of Legendre's polynomials while the second was Functions of bounded variation.