 
According to algebraic
theory systems of perpendicular (orthogonal) polynomials f_{i} can be
constructed, that span a so called function space.
Orthogonality has the meaning of being zero of the inproducts <f_{i},
f_{j}>
for i <> j. This function inproduct is defined as above mentioned integral, where
w(x) is
called the weight function.
Always true is that f_{i} is a polynomial of degree i with the remarkable quality
that all points of intersection with the xaxis are real and different, and are in the
interval [a, b]. Some orthogonal polynomials have been given special attention, and were
named to their explorer:





 polynomial 
interval 
w(x) 
f_{n}(x) 
abbreviation 
Hermite 
[∞,∞] 


H_{n} (x) 
Laguerre 
[0,∞] 


L_{n} (x) 
Legendre 
[1 , 1] 
1 

P_{n} (x) 
Tchebyscheff ^{1)} 
[1 , 1] 

?? 
T_{n} (x) 
The polynomials are
also solutions of differential equations with the same name (equation of Hermite, Laguerre
and so on), with whom I don't want you to torture.
The polynomials of Legendre are also called the spherical
functions of the first kind. A spherical function is a solution of the
equation of Laplace ^{2)}, an equation to
which a lot of theoretical problems in physics and astronomy can be reduced. The spherical term
is derived from the solution method, where variables are split in spherical coordinates.
Working out this solution, derivatives of polynomials of Legendre, as function of cosφ, are formed.
These are called the associated polynomials of Legendre^{3)}.
From the Legendre polyonomial the Legendre trigonometric can be derived.
notes
1) Or: Tsjebysjev
2) Equation of Laplace: Δf = 0
3) See WolframMathWorld
