Page 308 - Schaum's Outline of Differential Equations

CHAP. 29] SOME CLASSICAL DIFFERENTIAL EQUATIONS 291

These polynomials enjoy many properties, orthogonality being one of the most important. This condition,
which is expressed in terms of an integral, makes it possible for "more complicated" functions to be expressed
in terms of these polynomials, much like the expansions which will be addressed in Chapter 33. We say that the
polynomials are orthogonal with respect to a weight function (see, for example, Problem 29.2).
We now list the first five polynomials (n = 0, 1, 2, 3, 4) of each type:

• Chebyshev Polynomials, T n(x):
T 0(x) = 1

T l(x)=x
2
T 2(x) = 2x -l
T 3(x) = 4x*-3x
2
4
T 4(x) = 8x - &C + 1

Hermite Polynomials, n(x):
H
H 0(x)=l
H!(X) = 2x
2
H 2(x) = 4x -2
3
H 3(x) = 8.x - I2x
2
4
H 4(x) = I6x - 48.x + 12

Laguerre Polynomials,(x): L
n
L 0(x) = 1
L v(x) = -x + 1
2
L 2(x) =x -4x+2
2
L 3(x) = -x* + 9x -18^ + 6
4
L 4(x) =x - 16X 3 + 72X 2 - 96x + 24

Legendre Polynomials, n(x):
P
P 0(x) = 1

Pi(x) = x