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The first six Legendre polynomials. In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as ...
Associated Legendre functions for m = 0 Associated Legendre functions for m = 1 Associated Legendre functions for m = 2. The first few associated Legendre functions, including those for negative values of m, are: =
The general Legendre equation reads ″ ′ + [(+)] =, where the numbers λ and μ may be complex, and are called the degree and order of the relevant function, respectively. . The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre polynomials P n; and when λ is an integer (denoted n), and μ = m is also an integer with | m | < n are the associated Legendre ...
The following proof shows that the polynomials obtained from the Rodrigues' formula obey the second order differential equation just given. This proof repeatedly uses the fact that the second derivative of B(x) and the first derivative of A(x) are constants.
Classical orthogonal polynomials appeared in the early 19th century in the works of Adrien-Marie Legendre, who introduced the Legendre polynomials. In the late 19th century, the study of continued fractions to solve the moment problem by P. L. Chebyshev and then A.A. Markov and T.J. Stieltjes led to the general notion of orthogonal polynomials.
The first few Chebyshev polynomials of the first kind in the domain −1 < x < 1: The flat T 0, T 1, T 2, T 3, T 4 and T 5. ... Legendre polynomials; Laguerre ...
Adrien-Marie Legendre (/ l ə ˈ ʒ ɑː n d ər,-ˈ ʒ ɑː n d /; [3] French: [adʁiɛ̃ maʁi ləʒɑ̃dʁ]; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him.
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by Pál Turán () (and first published by Szegö (1948)).There are many generalizations to other polynomials, often called Turán's inequalities, given by (E. F. Beckenbach, W. Seidel & Otto Szász 1951) and other authors.