Legendre and Jacobi polynomials - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Algebra : Polynomials : Orthogonal Polynomials : orthopoly/P

orthopoly[P] - Legendre and Jacobi polynomials

Calling Sequence

P(n, a, b, x)

P(n, x)

Parameters

n

-

non-negative integer

x

-

algebraic expression

a, b

-

rational numbers greater than -1 or nonrational algebraic expressions

Description

• 

The P(n, a, b, x) function computes the nth Jacobi polynomial with parameters a and b evaluated at x.

  

In the case of only two arguments, P(n, x) computes the nth Legendre (spherical) polynomial which is equal to P(n, 0, 0, x).

• 

These polynomials are orthogonal on the interval 1,1 with respect to the weight function wx=1xa1+xb when a and b are greater than -1. They satisfy:

−11wtPm,a,b,tPn,a,b,tⅆt=0nm2a+b+1Γa+n+1Γn+b+12n+a+b+1n!Γn+a+b+1n=m

  

The Jacobi polynomials are undefined for negative integer values of a or b.

• 

Jacobi polynomials satisfy the following recurrence relation:

P0,a,b,x=1,

P1,a,b,x=12a12b+1+12a+12bx,

Pn,a,b,x=122n+a+b1a2b2+2n+a+b22n+a+bxPn1,a,b,xnn+a+b2n+a+b2n+a1n+b12n+a+bPn2,a,b,xnn+a+b2n+a+b2,for n>1.

Examples

withorthopoly:

P3,x

52x332x

(1)

P30,13

18024734042221205891132094649

(2)

P4,1,34,x

1154+1354x+418564x12+488251024x13+38083532768x14

(3)

P7,23,74,12

725899033738654705664

(4)

See Also

GAMMA, JacobiP, Legendre, numtheory[jacobi], numtheory[legendre]


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam